#411256

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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)(7,12)(8,11)(9,10)(14,17)(18,19)(21,22,24,28,25,27,33,35)(23,29,31,34,30,36,32,26);(1,2,4,3)(6,10)(11,12)(13,14,16,17)(15,19,20,18)(21,23)(22,26)(24,32)(25,30)(27,34)(28,36)(29,35)(31,33);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(22,27)(23,30)(28,35)(31,32);(1,3,4,2)(5,8,9,7)(6,11,10,12)(22,27)(26,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fg,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318452);