#8696270

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	#54845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=g,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, 8603466);