#6672368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579564);