#8492781

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	#40170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,13)(14,15)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(1,3,7,4)(2,8,11,9)(5,14,16,12)(6,13,10,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(35,44)(37,45)(40,48)(41,46);(1,3,7,4)(2,9,11,8)(5,12,16,14)(6,13,10,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,4,7,3)(2,9,11,8)(5,14,16,12)(6,13,10,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,5)(2,10)(3,12)(4,14)(6,11)(7,16)(8,13)(9,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,6,7,10)(2,5,11,16)(3,13,4,15)(8,14,9,12)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8399977);