#6373129

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,22,18,21,29,36)(15,19,25,32,23,30,37,41)(16,20,27,35,28,34,40,44)(24,31,38,42,26,33,39,43);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,26)(17,25)(18,23)(20,33)(21,30)(22,32)(24,28)(27,39)(29,37)(31,34)(35,43)(36,41)(38,40)(42,44);(2,4)(15,23)(19,30)(24,26)(25,37)(31,33)(32,41)(38,39)(42,43);(5,8,9,7)(6,10,11,12)(13,16,17,27,18,28,29,40)(14,20,22,35,21,34,36,44)(15,24,25,38,23,26,37,39)(19,31,32,42,30,33,41,43);(14,21)(19,30)(20,34)(22,36)(31,33)(32,41)(35,44)(42,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=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, 6280325);