#10463524

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

5429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,27,29)(20,25,30,32)(21,26,31,28);(9,10,12,14)(11,13,16,15)(18,23)(19,27)(25,32)(26,28);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16);(3,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32);(1,3)(2,5)(4,7)(6,8)(17,20)(18,25)(19,28)(21,24)(22,30)(23,32)(26,27)(29,31);(1,2)(3,5)(4,6)(7,8)(17,21)(18,26)(19,25)(20,29)(22,31)(23,28)(24,30)(27,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,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 subgroup of	#72336977
maximal normal subgroup of	#72336977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10370720);