#8695250

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

3937 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8602446);