#232126

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139322);