#495535

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=f,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313392, #72313451, #72313516, …
maximal normal subgroup of	#72313392, #72313451, #72313516, …
maximal quotient of	#72343684

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402731);