#229747

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

681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,d^c=di,f^c=fh

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, 136943);