#5021515

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219276, #72219277
maximal normal subgroup of	#72219276, #72219277

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4928711);