#496096

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ci,d^c=di

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, 403292);