#361402

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ei,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,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, 268598);