#7156037

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063233);