#6374373

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2908 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(22,26)(23,29)(25,32)(27,35)(30,36)(31,33);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,22)(23,30)(24,26)(25,28)(27,31)(29,35)(32,34)(33,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23,24,31)(22,27,26,30)(25,33,32,29)(28,35,34,36);(6,12)(10,13)(11,20)(18,19)(21,22)(23,27)(24,26)(25,28)(29,36)(30,31)(32,34)(33,35);(22,26)(27,30)(28,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281569);