#193446

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^b=eh,g^b=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320145, #72320207, #72320224
maximal normal subgroup of	#72320145, #72320207, #72320224
maximal quotient of	#72321942

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100642);