#8437753

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,10,14,8)(4,6,5,15)(11,16,12,13)(18,24)(19,20)(26,30)(27,29)(33,34,39,38)(35,43,44,40)(36,41,46,48)(37,42,45,47);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(35,44)(36,46)(40,43)(41,48);(1,4,7,5)(2,6,9,15)(3,11,14,12)(8,13,10,16)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,35,39,44)(34,40,38,43)(36,45,46,37)(41,42,48,47);(1,5,7,4)(2,6,9,15)(3,12,14,11)(8,13,10,16)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,36)(34,41)(35,45)(37,44)(38,48)(39,46)(40,42)(43,47);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28)(33,37,39,45)(34,42,38,47)(35,36,44,46)(40,48,43,41);(1,6)(2,5)(3,13)(4,9)(7,15)(8,12)(10,11)(14,16)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,38,39,34)(35,43,44,40)(36,41,46,48)(37,47,45,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=g,e^2=h,f^2=h,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, 8344949);