#8109396

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

4930 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,f^2=h,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 normal subgroup of	#72219279, #72219281
maximal quotient of	#72219274

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8016592);