#10460725

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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)(3,11,13,8)(5,9,12,14)(6,10,16,15)(18,22)(20,28)(23,25)(26,29)(33,34)(35,39)(36,45)(37,41)(38,42)(40,47)(43,44)(46,48);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,15,14,10)(18,22)(23,25)(24,30)(31,32)(33,35,38,44)(34,39,42,43)(36,46,47,37)(40,41,45,48);(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,47)(46,48);(1,4,7,2)(3,8,13,11)(5,14,12,9)(6,15,16,10)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36)(34,40)(35,37)(38,47)(39,48)(41,43)(42,45)(44,46);(1,5)(2,9)(3,6)(4,14)(7,12)(8,15)(10,11)(13,16)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,46)(43,44)(45,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,9)(11,14)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,41)(35,36)(38,46)(39,45)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367921);