#159428

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

4040 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328411, #72328434, #72328580, …
maximal normal subgroup of	#72328411, #72328434, #72328580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 66624);