#10460730

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^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.

Relationships

maximal quotient of

#72228716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367926);