#9498856

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

5196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265955, #72265959, #72265963
maximal normal subgroup of	#72265955, #72265959, #72265963
maximal quotient of	#72266543, #72266563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9406052);