#6372056

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5382 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,h^2=i,b^a=bf,c^a=cg,e^d=ei

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279252);