#201624

order	512
nilpotency class	3
isoclinism	isoclinic class of #201404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2617	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1439 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)(12,23,25,16)(13,26)(14,17)(15,21)(18,28)(20,33,36,40)(22,34)(24,31,27,32)(29,35,39,37)(30,38);(2,6,8,7)(9,11,13,21,14,22,28,38)(10,15,18,19,17,30,26,34)(12,20,27,37,25,36,24,35)(16,29,32,33,23,39,31,40);(9,12)(10,16)(11,20)(13,27)(14,25)(15,29)(17,23)(18,32)(19,33)(21,37)(22,36)(24,28)(26,31)(30,39)(34,40)(35,38);(10,17)(15,30)(16,23)(18,26)(19,34)(29,39)(31,32)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281986, #72340322, #72340404, …
maximal normal subgroup of	#72281986, #72340322, #72340404, …
maximal quotient of	#72282737

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108820);