#402858

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22,24,28,26,30,25,29)(23,31,33,35,32,36,34,27);(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,32)(22,27,30,35)(24,34,25,33)(28,36,29,31);(15,19)(17,20)(21,22,26,30)(23,27,32,35)(24,29,25,28)(31,33,36,34);(2,4)(5,7,8,11)(6,10,9,12)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

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

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, 310054);