#401627

order	512
nilpotency class	3
isoclinism	isoclinic class of #401626
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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1808 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)(5,6,10,14,11,16,9,15)(7,17,20,28,21,32,19,12)(8,22,25,13,26,30,24,31)(18,27,23,29,35,36,34,33);(5,7,9,19,11,21,10,20)(6,12,14,17,16,28,15,32)(8,23,24,18,26,34,25,35)(13,29,30,36,31,33,22,27);(5,8,11,26)(6,13,16,31)(7,18,21,35)(9,25,10,24)(12,27,28,36)(14,22,15,30)(17,33,32,29)(19,23,20,34);(2,4)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,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, 308823);