#8097025

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	#45530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=gh,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, 8004221);