#8926619

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

3765 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=gh,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, 8833815);