#8108711

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(36,43)(40,45)(42,47)(46,48);(2,6)(7,9)(8,12)(14,15)(19,26)(20,25)(21,30)(23,29)(34,38)(35,41)(40,45)(42,47);(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015907);