#8926634

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833830);