#8455656

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3,8,6)(2,9,7,11)(4,12,5,13)(10,15,14,16)(20,28)(25,30)(27,29)(31,32)(34,39)(35,42)(41,44)(43,48);(1,4,8,5)(2,10,7,14)(3,12,6,13)(9,15,11,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,5,8,4)(2,10,7,14)(3,13,6,12)(9,15,11,16)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,8,3)(2,11,7,9)(4,13,5,12)(10,16,14,15)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31);(1,7,8,2)(3,11,6,9)(4,14,5,10)(12,16,13,15)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 8362852);