#8430753

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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)(3,11)(4,15)(5,16)(6,10)(7,14)(8,13)(9,12)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,30)(31,32)(36,44)(40,47)(42,48)(45,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,19,22,29)(18,23,27,24)(20,26,30,32)(21,28,31,25)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,19)(18,24)(20,26)(21,28)(22,29)(23,27)(25,31)(30,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,13)(2,8,10,15)(3,12,14,5)(7,9,11,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,27)(28,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,8,11,15)(17,20,22,30)(18,25,27,28)(19,26,29,32)(21,24,31,23)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,2)(3,7)(4,15)(5,9)(6,10)(8,13)(11,14)(12,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,30)(25,29)(27,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337949);