#8110604

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

5084 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239253, #72239255
maximal normal subgroup of	#72239253, #72239255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017800);