#7302514

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7209710);