#893339

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267626
maximal normal subgroup of	#72267626
maximal quotient of	#72267682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 800535);