#6438528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	#34737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2477 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283249, #72283426, #72285023
maximal normal subgroup of	#72283249, #72283426, #72285023
maximal quotient of	#72319993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345724);