#6372007

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(5,6,7,8)(9,10)(11,17)(12,24)(13,18)(14,29)(15,20)(16,22)(19,25)(21,31)(23,32)(26,37)(27,33)(28,39)(30,36)(34,38)(35,40);(1,2,3,4)(6,8)(9,11,14,22,15,23,25,12)(10,16,19,17,20,24,29,32)(13,21,28,36,27,35,38,26)(18,30,34,31,33,37,39,40);(9,12,25,23,15,22,14,11)(10,17,29,16,20,32,19,24)(13,26,38,35,27,36,28,21)(18,31,39,30,33,40,34,37);(9,13)(10,18)(11,21)(12,26)(14,28)(15,27)(16,30)(17,31)(19,34)(20,33)(22,36)(23,35)(24,37)(25,38)(29,39)(32,40);(13,27)(18,33)(21,35)(26,36)(28,38)(30,37)(31,40)(34,39)

Polycyclic presentation:

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

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, 6279203);