#8926654

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=g,f^2=h,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, 8833850);