#8898927

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(3,12)(4,15)(5,6)(7,9)(8,14)(10,13)(11,16)(17,18,22,25)(19,23,29,26)(20,21,24,30)(27,28,31,32)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,7,14)(2,8,9,12)(4,11,13,5)(6,10,16,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(37,45)(40,44)(43,47)(46,48);(2,9)(6,16)(8,12)(10,15)(17,18,22,25)(19,26,29,23)(20,21,24,30)(27,32,31,28)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4)(2,10)(3,11)(5,14)(6,12)(7,13)(8,16)(9,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,5,7,11)(2,6,9,16)(3,13,14,4)(8,15,12,10)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,47)(42,48);(1,6)(2,11)(3,10)(4,12)(5,9)(7,16)(8,13)(14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806123);