#414501

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

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

1018 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)(7,12)(8,10)(9,11)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,33,36,32);(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,34)(22,27,30,36)(24,33,25,32)(28,35,29,31);(5,8,9,7)(6,10,11,12)(14,17)(18,20)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(1,2,3,4)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=efi,c^2=ef,d^2=gh,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, 321697);