#381672

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,19,22,31)(12,16)(13,17)(14,18)(15,21,27,25)(20,33,35,39)(23,29)(24,30)(26,32)(28,34,38,36)(37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,31)(18,27)(19,32)(21,26)(23,36)(24,35)(29,39)(30,38)(33,40)(34,37);(2,6)(5,8)(12,23)(13,26)(15,27)(16,29)(17,32)(20,34)(21,25)(24,37)(28,39)(30,40)(33,38)(35,36);(9,12,13,23,14,24,26,37)(10,16,17,29,18,30,32,40)(11,20,25,36,22,35,21,34)(15,28,31,39,27,38,19,33)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=eh,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226669, #72226672, #72226676
maximal normal subgroup of	#72226669, #72226672, #72226676

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288868);