#6438207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2920 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,26)(23,27)(24,28);(6,9)(10,12)(13,15,17,23)(14,18,19,27)(16,21,24,22)(20,25,28,26);(2,4)(5,7,8,11)(6,10,9,12)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(15,22)(16,24)(18,26)(20,28)(21,23)(25,27);(14,19)(18,27)(20,28)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312748, #72312818, #72312854
maximal normal subgroup of	#72312748, #72312818, #72312854
maximal quotient of	#72315795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345403);