#6454135

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	#24354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2401 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)(16,19)(17,20)(21,26)(22,27)(23,25)(24,28);(2,4)(13,15)(14,18)(16,22)(17,21)(19,27)(20,26)(23,24)(25,28);(6,9)(10,12)(14,19)(15,21)(17,24)(18,25)(22,23)(26,27);(15,21)(17,24)(18,26)(20,28)(22,23)(25,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319376, #72319519, #72319522
maximal normal subgroup of	#72319376, #72319519, #72319522

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361331);