#8932280

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839476);