#8712816

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^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, 8620012);