#354770

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1214 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)(7,12)(8,10)(9,11)(13,14,18,22,19,23,17,33)(15,24,28,34,25,39,27,20)(16,21,31,37,32,38,30,44)(26,40,42,36,29,43,41,35);(2,4)(13,15)(14,20,23,34)(16,29)(17,27)(18,28)(19,25)(21,36,38,35)(22,24,33,39)(26,32)(30,42)(31,41)(37,43,44,40);(5,7,9,8)(6,10,11,12)(15,25)(20,34)(24,39)(26,29)(27,28)(35,36)(40,43)(41,42);(5,8,9,7)(6,10,11,12)(13,16,18,31,19,32,17,30)(14,21,22,37,23,38,33,44)(15,26,28,42,25,29,27,41)(20,35,24,40,34,36,39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,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, 261966);