#10360913

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	#41058	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4212 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=h,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, 10268109);