#8202428

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	#39048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 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:

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

Polycyclic presentation:

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

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, 8109624);