#10423015

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	#48025	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(35,42)(39,46)(43,47)(44,48);(1,3,7,4)(2,8,10,5)(6,11,14,12)(9,15,16,13)(19,23)(20,27)(26,31)(28,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,7,3)(2,5,10,8)(6,11,14,12)(9,15,16,13)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5)(2,4)(3,10)(6,13)(7,8)(9,12)(11,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,6,7,14)(2,9,10,16)(3,11,4,12)(5,13,8,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(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,d^2=g,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, 10330211);