#8456962

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	#55028	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(18,24)(19,20)(26,31)(27,28)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,29)(30,31)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(17,19,23,20)(18,25,24,22)(21,27,30,28)(26,32,31,29)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,20,23,19)(18,25,24,22)(21,28,30,27)(26,32,31,29)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,21,23,30)(18,26,24,31)(19,27,20,28)(22,29,25,32)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,22,23,25)(18,20,24,19)(21,29,30,32)(26,28,31,27)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=h,e^2=hi,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, 8364158);