#8093212

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	#45801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

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

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, 8000408);