#6380889

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288085);