#992940

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(6,8)(10,14)(11,12)(17,18,21,23)(19,22,20,24);(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,22,23,24);(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,20,21,19)(18,22,23,24);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,15,14,16)(17,18)(19,24)(20,22)(21,23);(9,10)(11,16)(12,15)(13,14)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=gh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 900136);