#8096838

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	#45530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(3,10)(4,13)(5,12)(6,8)(7,9)(11,15)(14,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(34,40)(36,41)(42,48)(43,46);(1,3,7,12)(2,5,9,10)(4,14,6,11)(8,16,13,15)(20,28)(24,29)(26,32)(30,31)(33,34,39,40)(35,41,45,36)(37,42,44,48)(38,43,47,46);(1,4,7,6)(2,8,9,13)(3,11,12,14)(5,15,10,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,45)(34,41,40,36)(37,38,44,47)(42,43,48,46);(1,5)(2,3)(4,15)(6,16)(7,10)(8,11)(9,12)(13,14)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36)(34,35)(37,46)(38,42)(39,41)(40,45)(43,44)(47,48);(1,6,7,4)(2,8,9,13)(3,11,12,14)(5,16,10,15)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,37)(34,42)(35,38)(36,46)(39,44)(40,48)(41,43)(45,47);(1,2)(3,5)(4,8)(6,13)(7,9)(10,12)(11,15)(14,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,38)(34,43)(35,44)(36,42)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004034);