#386917

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×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

890 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)(7,8)(9,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,29)(25,26)(28,32)(30,35)(31,33)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,20,19)(22,26)(23,30)(24,32)(27,34)(31,36)(33,35);(5,8,9,10)(6,7,11,12)(13,15,17,16)(14,19,20,18)(21,23,24,30,25,31,32,36)(22,27,29,35,26,33,28,34);(6,11)(7,12)(13,16,17,15)(14,18,20,19)(21,24,25,32)(22,28,26,29)(23,30,31,36)(27,34,33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=efgi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=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, 294113);