#185523

order	512
nilpotency class	3
isoclinism	isoclinic class of #185517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3542	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238943, #72239065
maximal normal subgroup of	#72238943, #72239065
maximal quotient of	#72238963, #72239147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92719);