#8324630

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(3,6,9,15)(4,5,8,14)(11,12,13,16)(17,18,24,28)(19,29,30,23)(20,26,21,25)(22,32,31,27)(34,40)(36,45)(41,42)(43,46);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,19,24,30)(18,23,28,29)(20,22,21,31)(25,32,26,27)(33,34)(35,41)(36,38)(37,42)(39,40)(43,47)(44,46)(45,48);(3,9)(6,15)(11,13)(12,16)(17,20,24,21)(18,25,28,26)(19,22,30,31)(23,32,29,27)(33,35,39,37)(34,41,40,42)(36,43,45,46)(38,47,48,44);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,21,24,20)(18,26,28,25)(19,22,30,31)(23,32,29,27)(33,36)(34,38)(35,43)(37,46)(39,45)(40,48)(41,47)(42,44);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,22)(18,27)(19,21)(20,30)(23,25)(24,31)(26,29)(28,32)(33,37,39,35)(34,42,40,41)(36,43,45,46)(38,47,48,44);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,23,24,29)(18,19,28,30)(20,27,21,32)(22,25,31,26)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231826);