#8810108

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	#45536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(3,8)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,31,29)(33,34,40,46)(35,43,37,41)(36,45,39,42)(38,44,48,47);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,22,24)(18,23,26,20)(21,27,30,31)(25,29,32,28)(33,35,40,37)(34,41,46,43)(36,47,39,44)(38,45,48,42);(2,9)(3,4)(5,11)(13,15)(17,20,22,23)(18,24,26,19)(21,29,30,28)(25,27,32,31)(33,36,40,39)(34,42,46,45)(35,47,37,44)(38,43,48,41);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,37,40,35)(34,43,46,41)(36,47,39,44)(38,45,48,42);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38,40,48)(34,44,46,47)(35,42,37,45)(36,41,39,43);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,39,40,36)(34,45,46,42)(35,47,37,44)(38,43,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717304);