#189990

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1242 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,g^b=gh,g^c=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210843, #72211060
maximal normal subgroup of	#72210843, #72211060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97186);