#8534933

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	#57161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3)(2,9)(4,14)(5,11)(6,10)(7,15)(8,16)(12,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,4,8,5)(2,10,12,7)(3,14,16,11)(6,13,15,9)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,5,8,4)(2,10,12,7)(3,11,16,14)(6,13,15,9)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,6,8,15)(2,11,12,14)(3,10,16,7)(4,9,5,13)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,7,8,10)(2,5,12,4)(3,15,16,6)(9,11,13,14)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442129);