#8679883

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

3937 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(17,18,23,26)(19,27,22,24)(20,21,31,25)(28,32,29,30)(33,34)(35,41)(36,46)(37,43)(38,39)(40,44)(42,45)(47,48);(1,3,8,16)(2,9,12,13)(4,14,5,6)(7,15,10,11)(17,19,23,22)(18,24,26,27)(20,28,31,29)(21,30,25,32)(33,35,40,38)(34,41,44,39)(36,47,45,37)(42,43,46,48);(1,4,8,5)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,20)(18,25)(19,28)(21,26)(22,29)(23,31)(24,32)(27,30)(33,36)(34,42)(35,37)(38,47)(39,43)(40,45)(41,48)(44,46);(1,5,8,4)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,20)(18,25)(19,29)(21,26)(22,28)(23,31)(24,30)(27,32)(33,37,40,47)(34,43,44,48)(35,45,38,36)(39,46,41,42);(1,6)(2,11)(3,4)(5,16)(7,9)(8,14)(10,13)(12,15)(17,21,23,25)(18,20,26,31)(19,30,22,32)(24,28,27,29)(33,38,40,35)(34,41,44,39)(36,47,45,37)(42,48,46,43);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,13,14,9)(17,22,23,19)(18,24,26,27)(20,29,31,28)(21,30,25,32)(33,39,40,41)(34,38,44,35)(36,43,45,48)(37,42,47,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587079);