#10055458

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

5547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342294, #72342327, #72342345
maximal normal subgroup of	#72342294, #72342327, #72342345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9962654);