#852775

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2066 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=h,e^2=f,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 759971);