#8471018

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	#56438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=hi,e^2=hi,f^2=i,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, 8378214);