#5083053

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4990249);