#8137079

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	#35071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044275);