#7148394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(7,11)(10,12)(13,14,19,24)(15,21,27,35)(16,22,30,41)(17,23,18,32)(20,31,36,26)(25,37,43,38)(28,44,42,34)(29,39,33,40);(6,9)(7,11)(13,15)(14,20)(16,28)(17,26)(18,31)(19,27)(21,32)(22,38)(23,35)(24,36)(25,29)(30,42)(33,43)(34,40)(37,41)(39,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,25)(17,31)(18,26)(19,27)(20,32)(22,37)(23,36)(24,35)(28,29)(30,43)(33,42)(34,40)(38,41)(39,44);(6,9)(10,12)(13,16,19,30)(14,22,24,41)(15,25,27,43)(17,29,18,33)(20,34,36,44)(21,37,35,38)(23,39,32,40)(26,42,31,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,32)(15,26,27,31)(16,29,30,33)(20,35,36,21)(22,39,41,40)(25,42,43,28)(34,38,44,37)

Polycyclic presentation:

2^9|a^2=gi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055590);