#2229233

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2704 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328607, #72328617, #72328716
maximal normal subgroup of	#72328607, #72328617, #72328716
maximal quotient of	#72330945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2136429);