#10424613

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	#40692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(2,5)(8,10)(9,13)(15,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(35,39)(36,43)(42,47)(44,48);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=gh,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, 10331809);