#8908524

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	#47995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8815720);