#8463859

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	#39996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(19,20)(23,24)(27,28)(30,31)(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)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(34,35)(39,41)(40,42)(46,47);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(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)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(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)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(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=ghi,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, 8371055);