#8097727

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	#45738	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004923);