#6373532

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
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

2002 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)(9,10,15,20,16,21,14,32)(11,22,13,19,24,36,25,17)(12,26,30,35,31,40,29,18)(23,34,28,39,27,33,38,37);(2,5)(6,8)(9,11,14,25,16,24,15,13)(10,17,20,22,21,19,32,36)(12,27,29,28,31,23,30,38)(18,34,26,39,35,33,40,37);(1,3,4,7)(2,6,5,8)(9,12,14,29,16,31,15,30)(10,18,20,26,21,35,32,40)(11,23,25,38,24,27,13,28)(17,33,22,37,19,34,36,39);(9,13,15,24,16,25,14,11)(10,19,20,36,21,17,32,22)(12,28,30,27,31,38,29,23)(18,34,26,39,35,33,40,37);(11,24)(13,25)(17,19)(22,36)(23,27)(28,38)(33,34)(37,39)

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,c^b=ci,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239527, #72307191, #72308712
maximal normal subgroup of	#72239527, #72307191, #72308712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280728);