#229813

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

(1,2,4,3)(5,6,10,14)(7,15,16,20)(8,13,9,12)(11,18,19,17)(21,22,26,28,25,29,24,33)(23,27,31,36,32,30,34,35);(2,3)(7,16)(11,19)(15,20)(17,18)(21,22,24,33,25,29,26,28)(23,30,34,36,32,27,31,35);(5,7,10,16)(6,11,14,19)(8,17,9,18)(12,15,13,20)(21,23)(22,27)(24,34)(25,32)(26,31)(28,36)(29,30)(33,35);(1,3,4,2)(5,8,10,9)(6,12,14,13)(7,17,16,18)(11,15,19,20)(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=egh,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=de,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137009);