#7301008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,21,26)(19,28,31,23)(20,25,33,43)(22,36,35,27)(24,37,42,34)(29,39,30,44)(32,40,48,45)(38,46,41,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,31)(18,23,26,28)(20,30,33,29)(22,32,35,48)(24,41,42,38)(25,39,43,44)(27,40,36,45)(34,47,37,46);(9,11)(10,13)(12,16)(14,15)(17,20,21,33)(18,24,26,42)(19,29,31,30)(22,37,35,34)(23,38,28,41)(25,27,43,36)(32,47,48,46)(39,45,44,40);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,30)(21,33)(22,34)(23,39)(24,36)(26,43)(27,42)(28,44)(29,31)(32,47)(35,37)(38,40)(41,45)(46,48);(2,6)(5,7)(10,14)(13,15)(18,26)(19,31)(20,33)(27,36)(32,48)(34,37)(38,41)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7208204);