#495033

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,17,11,32)(7,13,21,36,19,34,22,50)(8,23,26,52,28,53,27,56)(9,29,30,40,31,57,24,15)(14,37,38,55,39,54,51,25)(18,41,45,64,43,62,46,67)(20,48,49,61,47,68,42,35)(33,58,60,66,59,65,63,44);(2,4)(5,7,10,21,12,19,11,22)(6,13,16,36,17,34,32,50)(8,18,26,45,28,43,27,46)(9,20,30,49,31,47,24,42)(14,33,38,60,39,59,51,63)(15,35,29,48,40,61,57,68)(23,41,52,64,53,62,56,67)(25,44,37,58,55,66,54,65);(5,8,24,54,11,27,31,55,12,28,30,37,10,26,9,25)(6,14,29,52,16,38,40,53,17,39,57,56,32,51,15,23)(7,18,42,65,22,46,47,66,19,43,49,58,21,45,20,44)(13,33,48,64,36,60,61,62,34,59,68,67,50,63,35,41);(7,19)(13,34)(18,43)(20,47)(21,22)(33,59)(35,61)(36,50)(41,62)(42,49)(44,66)(45,46)(48,68)(58,65)(60,63)(64,67)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402229);