#414469

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
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

1080 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)(21,22,25,28)(23,33,34,30)(24,36,35,29)(26,32,27,31);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(22,28)(29,36)(30,33)(31,32);(1,2,3,4)(14,17)(18,20)(21,23,26,35,25,34,27,24)(22,29,31,33,28,36,32,30);(13,15,16,19)(14,18,17,20)(21,24,27,34,25,35,26,23)(22,30,32,36,28,33,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321665);