#494831

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402027);