#7135117

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(17,18,23,25,22,28,21,27)(19,24,35,37,31,40,30,39)(20,26,36,41,33,43,34,44)(29,38,46,47,32,42,45,48);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,22,31)(18,24,28,40)(20,32,33,29)(21,30,23,35)(25,37,27,39)(26,42,43,38)(34,46,36,45)(41,48,44,47);(9,11,14,10)(12,15,13,16)(18,25)(21,23)(24,37)(26,41)(27,28)(30,35)(34,36)(38,47)(39,40)(42,48)(43,44)(45,46);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,22,33)(18,26,28,43)(19,29,31,32)(21,34,23,36)(24,38,40,42)(25,41,27,44)(30,45,35,46)(37,47,39,48);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(20,33)(26,43)(29,32)(34,36)(38,42)(41,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=hi,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, 7042313);