#8762049

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	#56339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=g,e^2=g,f^2=g,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, 8669245);