#829491

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1991 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)(3,4)(5,8)(6,7)(9,10,13,15)(11,16,12,14)(18,22)(19,24)(25,26,29,31)(27,30,32,28);(2,7)(3,5)(9,10)(11,14)(12,16)(13,15)(25,27,29,32)(26,28,31,30);(1,3,6,5)(2,8,7,4)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,28,29,30)(26,27,31,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,19,21,24)(18,23,22,20)(25,26)(27,28)(29,31)(30,32);(1,5,6,3)(2,8,7,4)(9,12,13,11)(10,14,15,16)(17,20,21,23)(18,24,22,19)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,c^2=f,d^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 736687);