#8712396

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

4176 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35,37,43)(34,38,40,41)(36,42,46,44)(39,47,45,48);(4,13)(7,12)(10,16)(14,15)(19,28)(20,29)(24,30)(25,32)(33,35,37,43)(34,38,40,41)(36,42,46,44)(39,47,45,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,36)(34,39)(35,42)(37,46)(38,47)(40,45)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619592);