#395778

order	512
nilpotency class	3
isoclinism	isoclinic class of #395775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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)(5,6,10,15,11,16,9,14)(7,17,19,12,21,28,20,29)(8,22,24,30,25,31,26,13)(18,33,35,36,23,27,34,32);(5,7,11,21)(6,12,16,29)(8,23,25,18)(9,19,10,20)(13,27,30,33)(14,28,15,17)(22,36,31,32)(24,35,26,34);(5,8,9,24,11,25,10,26)(6,13,14,22,16,30,15,31)(7,18,20,35,21,23,19,34)(12,27,17,32,29,33,28,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,25,26)(12,28,29,17)(13,22,30,31)(18,34,23,35)(27,36,33,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 302974);