#8131181

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	#57201	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038377);