#8463771

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	#2671	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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(18,20)(23,26)(24,28)(30,32)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(19,25)(23,26)(27,31)(30,32)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=hi,f^2=ghi,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, 8370967);