#8174401

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081597);