#10409980

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

4264 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)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(2,5)(8,9)(10,13)(15,16)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10317176);