#6817530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1705 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:

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

Polycyclic presentation:

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

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, 6724726);