#6596070

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#447
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3461 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307620, #72307628, #72307944
maximal normal subgroup of	#72307620, #72307628, #72307944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503266);