#6399991

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307187);