#8384226

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	#50368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8291422);