#8435086

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	#56695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8342282);