#353751

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,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, 260947);