#412704

order	512
nilpotency class	3
isoclinism	isoclinic class of #395944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319900);