#155106

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(2,4)(9,10,12,11)(13,14,17,16)(15,19,20,18)(21,22,24,26)(23,25,27,28);(6,8)(14,16)(18,19)(21,22,24,26)(23,25,27,28);(5,6)(7,8)(9,11,12,10)(13,15,17,20)(14,18,16,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(13,16,17,14)(15,19,20,18)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=efi,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328505, #72328614, #72328774, …
maximal normal subgroup of	#72328505, #72328614, #72328774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62302);