#230644

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137840);