#392554

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,22)(12,28,25,34)(13,31,33,38)(14,20)(15,21)(16,26)(17,30,37,40)(18,32,39,24)(19,23)(27,35)(29,36);(1,3,5,4)(2,6,7,8)(9,11,23,29,15,27,14,26)(10,16,20,35,21,36,19,22)(12,24,40,33,25,32,30,13)(17,18,28,31,37,39,34,38);(1,4,5,3)(2,6,7,8)(9,12,15,25)(10,17)(11,24,27,32)(13,29,33,26)(14,30,23,40)(16,18)(19,34)(20,28)(21,37)(22,38)(31,35)(36,39);(9,13,14,32,15,33,23,24)(10,18,19,38,21,39,20,31)(11,25,26,40,27,12,29,30)(16,34,22,37,36,28,35,17)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,d^b=di,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240734, #72307318
maximal normal subgroup of	#72240734, #72307318

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299750);