#6372009

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
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

4982 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)(11,21)(12,16)(13,18)(14,28)(15,20)(17,23)(19,29)(22,35)(24,30)(25,31)(26,33)(27,39)(32,36)(34,38)(37,40);(1,2,3,4)(6,8)(9,11,14,12,15,23,29,25)(10,16,19,21,20,31,28,17)(13,22,27,24,26,36,38,37)(18,30,34,35,33,40,39,32);(9,12,14,23,15,25,29,11)(10,17,19,16,20,21,28,31)(13,24,27,36,26,37,38,22)(18,32,34,30,33,35,39,40);(9,13)(10,18)(11,22)(12,24)(14,27)(15,26)(16,30)(17,32)(19,34)(20,33)(21,35)(23,36)(25,37)(28,39)(29,38)(31,40);(13,26)(18,33)(22,36)(24,37)(27,38)(30,40)(32,35)(34,39)

Polycyclic presentation:

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

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, 6279205);