#235527

order	512
nilpotency class	3
isoclinism	isoclinic class of #235516
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1587 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:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281986, #72286597, #72286708, …
maximal normal subgroup of	#72281986, #72286597, #72286708, …
maximal quotient of	#72282719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142723);