#8475095

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	#56299	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382291);