#8425572

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

3941 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,24,23)(19,20,26,22)(21,29,32,25)(27,31,30,28)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(3,10)(6,7)(11,16)(14,15)(17,19)(18,22)(20,23)(21,30)(24,26)(25,28)(27,32)(29,31)(34,36)(35,41)(40,44)(43,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,20,24,22)(18,19,23,26)(21,28,32,31)(25,27,29,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,24,32)(18,25,23,29)(19,27,26,30)(20,28,22,31)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22,24,20)(18,26,23,19)(21,31,32,28)(25,30,29,27)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,23,24,18)(19,20,26,22)(21,25,32,29)(27,31,30,28)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,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, 8332768);