#427190

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

938 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)(5,6,9,8)(7,11,12,10)(13,14,17,19)(15,18,16,20)(21,22,24,27)(23,29,31,26)(25,28,33,36)(30,35,32,34);(5,7,9,12)(6,10,8,11)(13,15,17,16)(14,18,19,20)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(5,6,9,8)(7,10,12,11)(13,16,17,15)(14,18,19,20)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(2,4)(5,8,9,6)(7,10,12,11)(23,31)(26,29)(30,32)(34,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,c^2=efi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334386);