#6372004

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5062 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315547, #72315555, #72315593, …
maximal normal subgroup of	#72315547, #72315555, #72315593, …
maximal quotient of	#72316211

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279200);