#8728490

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,30,32,29)(33,34)(35,44)(36,41)(37,48)(38,43)(39,46)(40,47)(42,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,32)(30,31)(33,35)(34,39)(36,37)(38,46)(40,48)(41,42)(43,44)(45,47);(3,10)(6,7)(11,16)(14,15)(20,28)(24,31)(27,32)(29,30)(33,36,38,47)(34,40,43,41)(35,37,46,45)(39,48,44,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,30,23,29)(33,36)(34,41)(35,45)(37,46)(38,47)(39,48)(40,43)(42,44);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,37,38,45)(34,42,43,48)(35,36,46,47)(39,41,44,40);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,27)(28,30)(31,32)(33,34)(35,39)(36,40)(37,48)(38,43)(41,47)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635686);