#4035826

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3700 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)(5,6,8,10)(7,11,12,9)(14,18)(15,16)(21,22)(23,24)(25,26)(27,28);(7,12)(9,11)(13,14,17,18)(15,20,16,19)(22,26)(24,28);(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,19,18,20)(21,23,25,27)(22,24,26,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,17,15)(14,20,18,19)(21,24,25,28)(22,23,26,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,19)(17,18)(21,23)(22,24)(25,27)(26,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300342, #72300400, #72300422
maximal normal subgroup of	#72300342, #72300400, #72300422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3943022);