#8926704

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(3,7)(4,14)(5,9)(6,10)(8,12)(11,13)(15,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34,39,44)(35,43,38,40)(36,46,45,41)(37,42,47,48);(3,11)(5,15)(7,13)(9,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,35,39,38)(34,40,44,43)(36,47,45,37)(41,48,46,42);(1,3,6,11)(2,7,10,13)(4,5,12,15)(8,16,14,9)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,41)(35,37)(38,47)(39,45)(40,42)(43,48)(44,46);(1,4)(2,8)(3,5)(6,12)(7,16)(9,13)(10,14)(11,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,16)(13,14)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,39,47)(34,42,44,48)(35,45,38,36)(40,46,43,41);(1,2)(3,13)(4,8)(5,9)(6,10)(7,11)(12,14)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,39,35)(34,43,44,40)(36,47,45,37)(41,48,46,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gh,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, 8833900);