#8341629

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	#58489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

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

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, 8248825);