#9170474

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

6122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261997, #72262230, C₂×#9170474
maximal normal subgroup of	#72261997, #72262230, C₂×#9170474
maximal quotient of	#72293931, #72293935, #72293940, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9077670);