#7143145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1971 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)(13,14,19,23)(15,24,28,32)(16,21,30,37)(17,18,33,22)(20,34,26,27)(25,38,40,42)(29,31,43,36)(35,44,39,41);(6,9)(7,11)(13,15,19,28)(14,20,23,26)(16,25,30,40)(17,32,33,24)(18,34,22,27)(21,35,37,39)(29,42,43,38)(31,44,36,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,30)(14,21,23,37)(15,25,28,40)(17,29,33,43)(18,31,22,36)(20,35,26,39)(24,38,32,42)(27,41,34,44);(6,9)(10,12)(13,17,19,33)(14,22,23,18)(15,26,28,20)(16,29,30,43)(21,36,37,31)(24,34,32,27)(25,39,40,35)(38,44,42,41);(5,7,8,11)(6,10,9,12)(16,30)(21,37)(25,40)(29,43)(31,36)(35,39)(38,42)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050341);