#8445996

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=i,f^2=g,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, 8353192);