#357823

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255615, #72255617, #72255619, …
maximal normal subgroup of	#72255615, #72255617, #72255619, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265019);