#790115

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

2024 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fg,e^2=fg,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, 697311);