#6373528

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239527, #72307194, #72308721
maximal normal subgroup of	#72239527, #72307194, #72308721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280724);