#6439072

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307403, #72307466, #72307587
maximal normal subgroup of	#72307403, #72307466, #72307587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346268);