#8696022

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,6)(4,10)(5,14)(7,11)(8,12)(9,13)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,45)(36,48)(37,43)(38,42)(39,44)(40,47)(41,46);(1,3,8,13)(2,9,12,6)(4,7,5,15)(10,16,14,11)(19,28)(20,29)(24,30)(25,32)(33,35)(34,40)(36,37)(38,46)(39,47)(41,43)(42,48)(44,45);(1,4,8,5)(2,10,12,14)(3,7,13,15)(6,11,9,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,36,39,46)(34,41,44,48)(35,37,47,38)(40,43,45,42);(1,5,8,4)(2,10,12,14)(3,7,13,15)(6,16,9,11)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,37,39,38)(34,42,44,43)(35,36,47,46)(40,48,45,41);(1,6)(2,3)(4,11)(5,16)(7,10)(8,9)(12,13)(14,15)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,38,39,37)(34,43,44,42)(35,46,47,36)(40,41,45,48);(1,7)(2,11)(3,5)(4,13)(6,14)(8,15)(9,10)(12,16)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,34)(35,40)(36,41)(37,43)(38,42)(39,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,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, 8603218);