#110848

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#388
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215752, #72216045, #72216052, …
maximal normal subgroup of	#72215752, #72216045, #72216052, …
maximal quotient of	#72238899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18044);