#6378043

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
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	C₂×#121	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(23,27)(24,29)(25,28)(26,30)(31,35)(32,36)(33,34);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,31)(22,27,30,35)(24,32,25,33)(28,34,29,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,28)(24,25)(27,34)(29,30)(32,33)(35,36);(6,12)(10,13)(11,20)(18,19)(23,31)(27,35)(32,33)(34,36);(1,3,4,2)(21,24,26,25)(22,29,30,28)(23,32,31,33)(27,36,35,34)

Polycyclic presentation:

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

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, 6285239);