#387724

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,27)(25,29)(28,32)(30,33)(31,35)(34,36);(5,7,9,6)(8,11,12,10)(14,18)(15,16)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(5,8,9,12)(6,10,7,11)(13,15,17,16)(14,19,18,20)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(6,7)(10,11)(13,16,17,15)(14,19,18,20)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=e,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,f^c=fi,f^d=fi

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, 294920);