#8425622

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332818);