#9794270

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	#55601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,24)(19,26,28,23)(20,29,30,25)(21,31,32,27)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,28)(24,26)(29,32)(30,31)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,3)(7,9)(8,10)(14,15)(18,24)(20,30)(23,26)(27,31)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,30)(18,25,24,29)(19,27,28,31)(21,26,32,23)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,26)(20,27)(21,29)(22,28)(23,24)(25,32)(30,31)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,32)(18,27,24,31)(19,25,28,29)(20,26,30,23)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9701466);