#8517512

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	#53684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424708);