#407668

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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)(9,10,13,17,14,18,26,32)(11,19,21,27,22,31,25,15)(12,16,23,29,24,30,37,40)(20,33,34,38,35,39,36,28);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,31)(18,27)(19,32)(21,26)(23,36)(24,35)(29,39)(30,38)(33,40)(34,37);(15,27)(19,31)(28,38)(33,39);(2,6)(5,8)(9,12,13,23,14,24,26,37)(10,16,17,29,18,30,32,40)(11,20,25,36,22,35,21,34)(15,28,31,39,27,38,19,33)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^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	#72265390, #72307302, #72307382
maximal normal subgroup of	#72265390, #72307302, #72307382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314864);