#8922602

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

3912 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)(4,12)(5,11)(6,7)(8,13)(10,15)(14,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3,6,11)(2,5,7,9)(4,10,13,14)(8,15,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(35,37)(40,42)(43,44)(47,48);(2,7)(4,13)(5,9)(10,14)(19,25)(20,28)(23,27)(24,29)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,16)(11,14)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,13)(12,16)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,5,6,9)(2,3,7,11)(4,15,13,16)(8,10,12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,39,34)(35,40,37,42)(36,45,46,41)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829798);