#251962

order	512
nilpotency class	3
isoclinism	isoclinic class of #251935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#32889	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1522 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)(11,16)(12,23)(13,18)(14,29)(15,20)(17,27)(19,32)(21,37)(22,28)(24,40)(25,39)(26,36)(30,34)(31,35)(33,38);(1,3,4,7)(2,5,6,8)(9,11,15,18)(10,16,20,13)(12,24)(14,22,32,35)(17,33)(19,31,29,28)(21,26)(23,37)(25,34)(27,39)(30,38)(36,40);(5,8)(9,12)(10,17)(11,21)(13,25)(14,30)(15,26)(16,33)(18,24)(19,36)(20,34)(22,39)(23,29)(27,32)(28,37)(31,40)(35,38);(9,13,15,16)(10,18,20,11)(12,25,26,33)(14,31,32,28)(17,24,34,21)(19,22,29,35)(23,38,36,39)(27,37,30,40)

Polycyclic presentation:

2^9|b^2=ehi,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303845
maximal normal subgroup of	#72303845
maximal quotient of	#72304006, #72304007

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159158);