#522252

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3474 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=gh,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, 429448);