#10325389

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	#36839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232585);