#6465657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3246 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372853);