#355032

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,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, 262228);