#353606

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 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:

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

Polycyclic presentation:

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

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, 260802);