#401617

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1856 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)(7,12,18,27,19,28,25,32)(8,20,24,13,22,31,23,30)(17,29,21,35,33,36,34,26);(5,7)(6,12)(8,21,22,34)(9,25)(10,18)(11,19)(13,29,30,36)(14,32)(15,27)(16,28)(17,24,33,23)(20,26,31,35);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,26)(14,31)(15,20)(16,30)(18,34)(19,33)(21,25)(27,36)(28,35)(29,32);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257124, #72257147, #72257150
maximal normal subgroup of	#72257124, #72257147, #72257150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308813);