#507737

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265562, #72265585, #72265609
maximal normal subgroup of	#72265562, #72265585, #72265609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414933);