#8445246

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	#37570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

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

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, 8352442);