#7630938

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538134);