#411328

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 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)(13,14)(15,17)(16,18)(19,20)(21,22,26,29)(23,31,34,35)(24,27,25,30)(28,32,36,33);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,17,18,20)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(6,9)(10,12)(21,23,26,34)(22,28,29,36)(24,33,25,32)(27,35,30,31);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,29)(27,30)(28,36)(31,35)

Polycyclic presentation:

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

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, 318524);