#392549

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
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

720 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)(11,22,26,16)(12,27,30,35)(13,20)(14,21)(15,19)(17,28,34,29)(18,25,32,24)(23,33)(31,37)(36,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,31)(18,21)(19,22)(20,32)(27,40)(28,37)(29,38)(30,39)(33,35)(34,36);(1,4,5,3)(2,6,7,8)(9,12,13,28,15,30,14,29)(10,17,21,35,19,34,20,27)(11,23,24,37,26,39,25,38)(16,31,18,33,22,36,32,40);(9,11)(10,18,19,32)(12,23)(13,24)(14,25)(15,26)(16,21,22,20)(17,33,34,40)(27,31,35,36)(28,37)(29,38)(30,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299745);