#8455711

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(34,39)(35,42)(41,44)(43,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gh,e^2=hi,f^2=h,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, 8362907);