#8093563

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000759);