#8475140

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,26)(19,25,20,24)(21,31,32,22)(27,30,29,28)(36,44)(40,45)(42,48)(46,47);(2,7)(3,10)(9,12)(11,16)(17,19,23,20)(18,24,26,25)(21,29,32,27)(22,28,31,30)(33,34)(35,39)(36,45)(37,41)(38,43)(40,44)(42,46)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,23,20)(18,25,26,24)(21,27,32,29)(22,28,31,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,19)(18,25,26,24)(21,29,32,27)(22,28,31,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21)(18,22)(19,27)(20,29)(23,32)(24,30)(25,28)(26,31)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,29)(25,27)(26,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=h,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, 8382336);