#212299

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,24)(22,23)(25,26,29,28,30,32,38,37)(27,31,35,36,33,39,40,34);(2,5)(6,8)(9,11,13,21)(10,15,18,19)(12,22,14,20)(16,23,17,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,39)(34,37)(38,40);(1,3,4,7)(2,6,5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(27,33)(31,39)(34,36)(35,40);(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,32,30,37,38,26)(27,34,35,31,33,36,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eh,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dhi,d^c=dh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119495);