#315123

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222319);