#511885

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,11,16)(7,12,18,33)(8,21,23,54)(9,25,27,56)(10,30,28,15)(13,36,37,22)(14,39,41,29)(17,43,44,65)(19,49,47,67)(20,52,48,35)(24,55,31,40)(26,42,57,38)(32,58,59,45)(34,62,61,51)(46,66,53,63)(50,64,68,60);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,32)(14,34)(15,35)(16,33)(21,43)(22,45)(23,44)(24,46)(25,49)(26,50)(27,47)(28,48)(29,51)(30,52)(31,53)(36,58)(37,59)(38,60)(39,62)(40,63)(41,61)(42,64)(54,65)(55,66)(56,67)(57,68);(5,8)(6,13)(7,17)(9,26)(10,31)(11,23)(12,32)(14,40)(15,42)(16,37)(18,44)(19,50)(20,53)(21,25)(22,29)(24,28)(27,57)(30,38)(33,59)(34,63)(35,64)(36,39)(41,55)(43,49)(45,51)(46,48)(47,68)(52,60)(54,56)(58,62)(61,66)(65,67);(7,18)(9,27)(12,33)(14,41)(17,44)(20,48)(21,54)(22,36)(25,56)(26,57)(29,39)(32,59)(35,52)(40,55)(46,53)(60,64)

Polycyclic presentation:

2^9|a^2=i,b^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419081);