#400766

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250285, #72257156, #72257191, …
maximal normal subgroup of	#72250285, #72257156, #72257191, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307962);