#229941

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	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 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)(6,10,14,12)(8,16)(13,19,20,17)(15,18)(21,22,25,29,26,23,27,31)(24,28,33,35,32,30,34,36);(5,6,11,10,9,14,7,12)(8,17,18,20,16,19,15,13)(21,23,25,31,26,22,27,29)(24,30,33,36,32,28,34,35);(2,4)(5,7,9,11)(6,12,14,10)(8,18,16,15)(13,17,20,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8,11,18,9,16,7,15)(6,13,12,19,14,20,10,17)(24,32)(28,30)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 137137);