#498067

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=ghi,h^b=hi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309518, #72310202, #72310206, …
maximal normal subgroup of	#72309518, #72310202, #72310206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405263);