#478157

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
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

942 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)(5,6,8,11)(7,10,12,9)(13,14,18,15)(16,20,17,19)(21,22,25,28)(23,26,31,35)(24,27,32,36)(29,33,30,34);(5,7)(6,9)(8,12)(10,11)(13,15,18,14)(16,20,17,19)(21,23)(22,26)(24,29)(25,31)(27,33)(28,35)(30,32)(34,36);(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,19,15,20)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35);(2,4)(7,12)(9,10)(13,17,18,16)(14,19,15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385353);