#516809

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3138 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fi,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

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, 424005);