#184048

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286191, #72286355, #72322026, …
maximal normal subgroup of	#72286191, #72286355, #72322026, …
maximal quotient of	#72287085, #72319430

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91244);