#8107296

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	#41306	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

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

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, 8014492);