#8434758

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	#55627	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=gh,f^2=g,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, 8341954);