#8094653

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	#45641	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,24)(20,26)(21,29)(22,23)(25,30)(27,28)(31,32)(36,44)(40,45)(42,48)(46,47);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(34,38)(39,41)(40,45)(46,47);(5,13)(9,16)(11,12)(14,15)(18,23)(19,26)(25,29)(27,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,15,16,14)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001849);