#414484

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
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

1056 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fi,d^2=ghi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di

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, 321680);