#395582

order	512
nilpotency class	3
isoclinism	isoclinic class of #395581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2153 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:

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

Polycyclic presentation:

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

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, 302778);