#8696245

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=g,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, 8603441);