#247702

order	512
nilpotency class	3
isoclinism	isoclinic class of #247687
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1243 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fi,e^d=eh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154898);