#10457932

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	#52477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gh,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, 10365128);