#10521341

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	#51758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=ghi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428537);