#232125

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×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=eh,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, 139321);