#409853

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

1112 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fi,d^2=ef,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,d^c=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, 317049);