#7054841

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	#25115	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2106 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,20,25,37)(16,21,29,38)(17,30,33,23)(19,31,35,26)(24,36,41,43)(27,34,42,44)(28,40,32,39);(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,19,22,35)(16,27,29,42)(17,31,33,26)(20,23,37,30)(21,36,38,43)(24,28,41,32)(34,39,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,30)(21,36)(22,37)(23,35)(27,32)(28,42)(29,41)(31,33)(34,39)(38,43)(40,44);(7,12)(9,11)(13,16,18,29)(14,21,22,38)(15,24,25,41)(17,32,33,28)(19,34,35,44)(20,36,37,43)(23,40,30,39)(26,27,31,42);(6,10)(9,11)(14,22)(15,25)(21,38)(23,30)(24,41)(26,31)(27,42)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,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, 6962037);