#8436744

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	#56014	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,35)(36,43)(40,42)(44,45);(2,7)(3,10)(8,14)(11,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36)(34,39)(35,41)(37,45)(38,43)(40,48)(42,47)(44,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,e^2=gh,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, 8343940);