#8172667

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

5479 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342290, #72342324, #72342342
maximal normal subgroup of	#72342290, #72342324, #72342342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8079863);