#8932258

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	#53993	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=gh,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, 8839454);