#10444994

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10352190);