#8093417

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,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, 8000613);