#877565

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2562 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^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	#72268606
maximal normal subgroup of	#72268606
maximal quotient of	#72268609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 784761);