#6399122

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(23,26)(24,27)(25,28)(29,32)(30,35)(31,33)(34,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,25,30)(22,26,28,35)(24,31,32,36)(27,33,29,34);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(7,15)(10,13)(14,20)(17,19);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(22,28)(23,30)(27,29)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306318);