#8186977

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	#53095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(6,8)(7,11)(9,14)(10,15)(13,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29)(34,39)(35,42)(41,44)(43,48);(2,8)(5,7)(9,12)(13,15)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,6,14)(2,9,8,12)(3,10,11,16)(5,13,7,15)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,12,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,32)(30,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094173);