#8436827

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	#53930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8344023);