#8472813

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	#56788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380009);