#8810194

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	#44570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,e^2=h,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, 8717390);