#7148228

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	#45101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

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

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, 7055424);