#495112

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1955
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402308);