#8821711

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	#44032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8728907);