#703985

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15273	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2707 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=g,d^2=fgh,e^2=fgh,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	#72263269, #72263289, #72263305
maximal normal subgroup of	#72263269, #72263289, #72263305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 611181);