#6444389

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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

3390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

(2,4)(9,10,12,11,14,16,13,15);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15);(1,2,3,4)(6,8)(9,10,12,11,14,16,13,15);(6,8)(9,10,12,11,14,16,13,15);(9,12,14,13)(10,15,16,11)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281059, #72281173, #72281198, …
maximal normal subgroup of	#72281059, #72281173, #72281198, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351585);