#8463763

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(18,20)(23,26)(24,28)(30,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(19,25)(23,26)(27,31)(30,32)(34,35)(39,41)(40,42)(46,47);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 8370959);