#407057

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1265 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)(3,5)(4,6)(7,8)(9,10,13,17,14,18,20,28)(11,19,22,30,23,34,27,15)(12,24,25,38,26,16,31,33)(21,35,36,32,37,40,39,29);(1,3,4,7)(2,5,6,8)(9,11,20,22,14,23,13,27)(10,15,28,19,18,30,17,34)(12,21,31,36,26,37,25,39)(16,32,38,40,24,29,33,35);(9,12,20,31,14,26,13,25)(10,16,28,38,18,24,17,33)(11,21,22,36,23,37,27,39)(15,29,19,35,30,32,34,40);(2,6)(5,8)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306970, #72307350, #72307382
maximal normal subgroup of	#72306970, #72307350, #72307382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314253);