#10423007

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	#55293	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(18,23)(20,24)(25,31)(28,29)(35,42)(39,46)(43,47)(44,48);(3,7)(4,11)(10,15)(12,16)(17,18,22,23)(19,24,27,20)(21,25,30,31)(26,29,32,28)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,6,7)(2,4,9,11)(5,10,14,15)(8,12,13,16)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,29,31,28)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,2)(3,4)(5,13)(6,9)(7,11)(8,14)(10,16)(12,15)(17,20)(18,19)(21,29)(22,24)(23,27)(25,32)(26,31)(28,30)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,15)(10,13)(14,16)(17,18)(19,24)(20,27)(21,25)(22,23)(26,29)(28,32)(30,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,21,22,30)(18,25,23,31)(19,26,27,32)(20,28,24,29)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,f^2=g,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, 10330203);