#355821

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cei,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263017);