#7622670

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3475	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,e^d=ei

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321414, #72322062, #72322106
maximal normal subgroup of	#72321414, #72322062, #72322106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529866);