#8926623

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(35,41)(39,42)(43,47)(46,48);(2,4)(7,11)(8,12)(14,16)(18,23)(19,26)(24,29)(27,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(5,13)(8,12)(9,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=gh,f^2=g,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, 8833819);