#10422693

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	#39326	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(2,5)(8,10)(9,13)(15,16)(18,23)(20,25)(24,29)(28,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,31)(28,29)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=gh,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, 10329889);