#203685

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

668 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:

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

Polycyclic presentation:

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

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, 110881);