#6377762

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284958);