#8696056

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	#43649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(3,6)(4,10)(5,14)(7,11)(8,12)(9,13)(15,16)(19,21)(24,27)(28,30)(29,31)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,8,13)(2,9,12,6)(4,7,5,15)(10,16,14,11)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(35,44)(36,45)(40,46)(41,48);(1,4,8,5)(2,10,12,14)(3,7,13,15)(6,11,9,16)(17,19,23,21)(18,24,26,27)(20,28,25,30)(22,29,32,31)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,5,8,4)(2,10,12,14)(3,7,13,15)(6,16,9,11)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,32)(27,31)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,6)(2,3)(4,11)(5,16)(7,10)(8,9)(12,13)(14,15)(17,21,23,19)(18,24,26,27)(20,30,25,28)(22,29,32,31)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,7)(2,11)(3,5)(4,13)(6,14)(8,15)(9,10)(12,16)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603252);