#495529

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 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)(3,6)(4,5)(7,8)(9,10,13,18,14,19,28,30,15,20,12,25,27,34,26,16)(11,17,23,32,24,33,38,39,21,31,22,36,35,40,37,29);(10,16,20,30)(12,26)(13,28)(14,27)(17,29,31,39)(18,19,25,34)(22,37)(23,38)(24,35)(32,33,36,40);(2,5)(6,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,21)(16,29)(18,32)(19,33)(20,31)(25,36)(26,37)(27,35)(28,38)(30,39)(34,40);(1,3,4,7)(2,6,5,8)(11,21)(17,31)(22,23)(24,35)(29,39)(32,36)(33,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239524, #72239543, #72239558, …
maximal normal subgroup of	#72239524, #72239543, #72239558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402725);