#152297

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59493);