#829484

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

1999 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)(3,7)(4,8)(5,6)(10,11)(14,15)(17,18,22,23)(19,20,21,24)(26,30)(27,32);(2,6)(7,8)(9,10,13,11)(12,15,16,14)(18,23)(19,21);(1,3,5,4)(2,7,6,8)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,24,23,20)(25,26)(27,31)(28,32)(29,30);(1,4,5,3)(2,7,6,8)(9,11,13,10)(12,14,16,15)(17,20)(18,19)(21,23)(22,24)(25,27,29,32)(26,31,30,28);(1,2)(3,7)(4,8)(5,6)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,24,23,20)(25,28,29,31)(26,32,30,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=f,d^2=hi,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, 736680);