#794468

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fgh,d^2=h,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, 701664);