#714496

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2956 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,22)(23,24)(26,29)(30,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18,21,22)(19,20,23,24)(25,26)(27,31)(28,29)(30,32);(3,7)(6,8)(11,12)(14,16)(18,22)(19,23)(25,26,28,29)(27,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,14,15,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329089, #72329114, #72329190
maximal normal subgroup of	#72329089, #72329114, #72329190
maximal quotient of	#72330382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 621692);