#532546

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 439742);