#7054324

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,18,23)(16,27,30,33)(17,22,32,37)(21,28,36,25)(26,34,31,38)(29,40,42,43)(35,41,44,39);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,24,23)(16,28,30,25)(17,26,32,31)(21,27,36,33)(22,34,37,38)(29,41,42,39)(35,40,44,43);(5,7,8,12)(6,9,10,11)(13,16,19,30)(14,21,24,36)(15,25,18,28)(17,29,32,42)(20,33,23,27)(22,35,37,44)(26,39,31,41)(34,43,38,40);(2,4)(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,31)(19,32)(20,34)(21,35)(23,38)(24,37)(25,39)(27,40)(28,41)(30,42)(33,43)(36,44);(6,10)(9,11)(13,15,19,18)(14,23,24,20)(16,25,30,28)(17,31,32,26)(21,27,36,33)(22,34,37,38)(29,41,42,39)(35,43,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6961520);