#494069

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 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)(5,6,10,17,11,18,28,36,12,15,29,30,9,16,20,35)(7,19,23,34,24,31,26,33,21,13,8,25,22,32,27,14);(1,2,4,3)(5,7,9,22,12,21,11,24)(6,13,16,31,15,19,18,32)(8,20,26,29,23,28,27,10)(14,30,25,36,33,17,34,35);(1,3,4,2)(5,8)(6,14,15,33)(7,20)(9,26)(10,24)(11,27)(12,23)(13,30,19,17)(16,25,18,34)(21,28)(22,29)(31,36,32,35);(7,21)(8,23)(13,19)(14,33)(22,24)(25,34)(26,27)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,g^b=gi,d^c=di,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, 401265);