#355050

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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

868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=efh,e^2=i,f^2=i,h^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, 262246);