#8926665

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	#56558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,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, 8833861);