#8664356

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571552);