#8926597

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

3788 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=h,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, 8833793);