#7148229

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	#45196	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)(17,30,32,34,31,22,26,37)(28,41,39,35,42,40,44,43);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,24,23)(16,25,29,33)(17,26,31,32)(21,27,36,38)(22,34,30,37)(28,39,42,44)(35,40,43,41);(5,6)(7,10)(8,9)(11,12)(13,16,19,29)(14,21,24,36)(15,25,18,33)(17,28,31,42)(20,27,23,38)(22,35,30,43)(26,39,32,44)(34,40,37,41);(2,4)(6,9)(10,12)(13,17,19,31)(14,22,24,30)(15,26,18,32)(16,28,29,42)(20,34,23,37)(21,35,36,43)(25,39,33,44)(27,40,38,41);(5,7,8,11)(6,10,9,12)(17,31)(22,30)(26,32)(28,42)(34,37)(35,43)(39,44)(40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,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, 7055425);