#8436613

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=h,e^2=g,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, 8343809);