#8202493

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,39,42)(35,41,36,40)(37,47,48,38)(43,46,45,44);(2,6)(7,9)(8,13)(14,15)(19,27)(20,30)(23,32)(24,29)(33,35,39,36)(34,40,42,41)(37,45,48,43)(38,44,47,46);(4,12)(8,13)(10,16)(14,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31)(33,36,39,35)(34,41,42,40)(37,45,48,43)(38,44,47,46);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(33,37)(34,38)(35,43)(36,45)(39,48)(40,46)(41,44)(42,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37)(34,38)(35,43)(36,45)(39,48)(40,46)(41,44)(42,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38)(34,37)(35,44)(36,46)(39,47)(40,45)(41,43)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109689);