#8093591

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

4041 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)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31)(34,40)(35,38)(36,42)(45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,34,39,40)(35,43,38,41)(36,46,42,37)(44,48,47,45);(2,6)(7,9)(8,13)(14,15)(17,19,22,27)(18,20,25,23)(21,26,30,32)(24,28,29,31)(33,35,39,38)(34,41,40,43)(36,47,42,44)(37,45,46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,36,39,42)(34,37,40,46)(35,44,38,47)(41,48,43,45);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,46)(41,44)(43,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38,39,35)(34,41,40,43)(36,47,42,44)(37,48,46,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=hi,f^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, 8000787);