#7062505

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	#42643	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(13,14,19,21,17,23,18,24)(15,25,32,36,27,20,28,34)(16,29,33,38,30,22,31,37)(26,35,42,43,40,39,41,44);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,27)(14,20,23,25)(16,26,30,40)(18,28,19,32)(21,34,24,36)(22,35,29,39)(31,41,33,42)(37,43,38,44);(5,7,8,12)(6,9,10,11)(14,21)(18,19)(20,34)(22,37)(23,24)(25,36)(28,32)(29,38)(31,33)(35,43)(39,44)(41,42);(2,4)(7,12)(9,11)(13,16,17,30)(14,22,23,29)(15,26,27,40)(18,31,19,33)(20,35,25,39)(21,37,24,38)(28,41,32,42)(34,43,36,44);(6,10)(9,11)(16,30)(22,29)(26,40)(31,33)(35,39)(37,38)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 6969701);