#224688

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 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)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,34)(28,32)(29,37)(30,33)(31,36)(35,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(27,35)(31,39)(34,40)(36,38);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(12,19)(21,24)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,39)(34,37);(1,4,5,3)(2,7,6,8)(25,28,30,29)(26,32,33,37)(27,36,35,38)(31,40,39,34)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=ei,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131884);