#7142936

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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(15,20)(16,26)(17,29)(18,32)(19,23)(21,24)(22,31)(25,38)(27,41)(28,34)(30,35)(33,37)(36,42)(39,44)(40,43);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,25,31,30)(21,34,37,26)(22,35,29,38)(27,39,43,42)(36,41,44,40);(5,6)(7,10)(8,9)(11,12)(13,16,19,28)(14,21,23,37)(15,24,18,33)(17,27,31,43)(20,34,32,26)(22,36,29,44)(25,39,30,42)(35,41,38,40);(2,4)(6,9)(10,12)(13,17)(14,22)(15,25)(16,27)(18,30)(19,31)(20,35)(21,36)(23,29)(24,39)(26,40)(28,43)(32,38)(33,42)(34,41)(37,44);(5,7,8,11)(6,10,9,12)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,30,31,25)(21,34,37,26)(22,38,29,35)(27,42,43,39)(36,40,44,41)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,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, 7050132);