#494919

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 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)(5,6,11,18,13,19,12,35)(7,20,25,40,22,47,26,55)(8,27,30,57,32,58,31,42)(9,17,33,37,34,45,10,16)(14,36,41,54,39,24,48,23)(15,28,43,59,44,60,61,29)(21,38,52,64,49,63,53,66)(46,50,56,62,65,67,68,51);(5,7,9,23,11,25,33,36,13,22,34,54,12,26,10,24)(6,14,37,47,35,48,17,40,19,39,16,20,18,41,45,55)(8,21,28,50,30,52,59,62,32,49,60,67,31,53,29,51)(15,38,57,65,61,66,27,56,44,63,42,46,43,64,58,68);(2,4)(5,8,9,28,11,30,33,59,13,32,34,60,12,31,10,29)(6,15,16,42,35,61,45,58,19,44,37,57,18,43,17,27)(7,21,23,50,25,52,36,62,22,49,54,67,26,53,24,51)(14,38,20,46,48,66,55,68,39,63,47,65,41,64,40,56);(7,22)(14,39)(20,47)(21,49)(23,54)(24,36)(25,26)(38,63)(40,55)(41,48)(46,65)(50,67)(51,62)(52,53)(56,68)(64,66)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,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, 402115);