#111497

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#392
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1084 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216622
maximal normal subgroup of	#72216622
maximal quotient of	#72214224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18693);