#115323

order	512
nilpotency class	3
isoclinism	isoclinic class of #115318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22519);