#231495

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,e^b=eh,g^b=gh,d^c=dh

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, 138691);