#8184862

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	#51931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092058);