#110202

order	512
nilpotency class	3
isoclinism	isoclinic class of #110189
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224692, #72224701, #72224703, …
maximal normal subgroup of	#72224692, #72224701, #72224703, …
maximal quotient of	#72257731, #72257737

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17398);