#8924989

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832185);