#10368633

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	#55236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,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, 10275829);