#8728306

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8635502);