#829496

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14722	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26,30,29)(27,28,32,31);(2,6)(4,7)(9,11,13,16)(10,14,15,12)(18,21)(22,23)(25,27,30,32)(26,31,29,28);(17,19)(18,22)(20,24)(21,23)(25,28)(26,32)(27,29)(30,31);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,24)(21,22)(25,28)(26,27)(29,32)(30,31);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,11,15,16)(17,18)(19,22)(20,21)(23,24)(25,29,30,26)(27,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304052, #72304075, #72304113
maximal normal subgroup of	#72304052, #72304075, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 736692);