#8932325

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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)(19,26)(22,24)(27,32)(29,31)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,11)(2,8,10,15)(4,12,13,5)(6,9,14,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(3,11)(4,13)(6,14)(8,15)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4)(2,6)(3,12)(5,11)(7,13)(8,9)(10,14)(15,16)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,12)(2,9,10,16)(3,13,11,4)(6,15,14,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,4)(3,9)(5,15)(7,14)(8,12)(10,13)(11,16)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=gi,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, 8839521);