#4599109

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6,9,10)(7,12,11,8)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,28,24,25);(6,10)(7,11)(15,20)(17,19)(22,27)(23,24);(6,10)(8,12)(13,15)(14,17)(16,20)(18,19)(21,23,26,24)(22,25,27,28);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,17)(16,20)(18,19)(21,24,26,23)(22,25,27,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,23)(24,27)(26,28)

Polycyclic presentation:

2^9|a^2=g,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327039, #72327046, #72327050
maximal normal subgroup of	#72327039, #72327046, #72327050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4506305);