#1435458

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2610 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232691, #72232692, #72232709
maximal normal subgroup of	#72232691, #72232692, #72232709

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1342654);