#522705

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 429901);