#8184753

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	#57643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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)(3,4,8,11)(5,13,14,9)(10,16,15,12)(17,18)(19,20)(21,30)(22,25)(23,26)(24,27)(28,31)(29,32)(33,34)(35,41)(36,47)(37,43)(38,48)(39,44)(40,45)(42,46);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,19)(18,21)(20,27)(22,28)(23,29)(24,30)(25,26)(31,32)(33,35,40,38)(34,41,45,48)(36,39,46,43)(37,47,44,42);(2,6)(3,8)(9,13)(10,15)(17,20,24,21)(18,19,27,30)(22,31,29,26)(23,25,28,32)(33,36)(34,42)(35,39)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4,7,11)(2,8,6,3)(5,12,14,16)(9,15,13,10)(17,21,24,20)(18,19,27,30)(22,26,29,31)(23,25,28,32)(33,37,40,44)(34,43,45,39)(35,42,38,47)(36,41,46,48);(1,5,7,14)(2,9,6,13)(3,10,8,15)(4,12,11,16)(17,22,24,29)(18,25,27,32)(19,28,30,23)(20,31,21,26)(33,38,40,35)(34,41,45,48)(36,43,46,39)(37,47,44,42);(1,6,7,2)(3,11,8,4)(5,13,14,9)(10,16,15,12)(17,23,24,28)(18,26,27,31)(19,29,30,22)(20,32,21,25)(33,39,40,43)(34,44,45,37)(35,46,38,36)(41,42,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,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, 8091949);