#183344

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,d^a=dh,c^b=ci,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285399, #72286355, #72322139
maximal normal subgroup of	#72285399, #72286355, #72322139
maximal quotient of	#72285597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90540);