#229951

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 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)(21,22,26,28,25,29,24,32)(23,27,30,36,31,35,33,34);(5,6,8,17,9,12,15,10)(7,14,16,11,13,18,20,19)(21,22,24,32,25,29,26,28)(23,27,33,34,31,35,30,36);(2,4)(21,23)(22,27)(24,33)(25,31)(26,30)(28,36)(29,35)(32,34);(5,7,15,20,9,13,8,16)(6,11,17,18,12,19,10,14)(21,24,25,26)(22,28,29,32)(23,30,31,33)(27,34,35,36)

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,d^b=de,e^b=eh,g^b=gh,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, 137147);