#8472379

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	#54841	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379575);