#8924310

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

3995 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,23,25)(19,24,29,32)(20,26,30,31)(21,27,28,22);(2,7)(4,9)(8,10)(12,14)(17,19,23,29)(18,24,25,32)(20,28,30,21)(22,31,27,26)(33,34)(35,42)(36,40)(37,39)(38,41)(43,47)(44,46)(45,48);(2,7)(3,11)(5,15)(12,14)(18,25)(19,29)(20,30)(22,27)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,21,23,28)(18,27,25,22)(19,30,29,20)(24,31,32,26)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,22,23,27)(18,28,25,21)(19,31,29,26)(20,24,30,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8831506);