#229999

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

631 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dhi

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, 137195);