#8425578

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	#55424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332774);