#8187002

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(3,12)(4,7)(5,15)(6,10)(8,11)(9,14)(13,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(34,35)(36,43)(40,42)(44,48);(1,3,8,14)(2,9,11,12)(4,10,5,16)(6,7,13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34,38,35)(36,39,43,41)(37,40,45,42)(44,46,48,47);(1,4,8,5)(2,7,11,15)(3,10,14,16)(6,9,13,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,5,8,4)(2,7,11,15)(3,10,14,16)(6,12,13,9)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,6,8,13)(2,10,11,16)(3,7,14,15)(4,9,5,12)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,7)(2,4)(3,13)(5,11)(6,14)(8,15)(9,10)(12,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094198);