#8925881

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

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, 8833077);