#7135099

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7042295);