#152004

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59200);