#127700

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#333
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#333
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,d^b=dh,e^b=ei,g^b=gi,h^b=hi

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, 34896);