#193510

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	#34909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

(1,2)(3,4)(5,6,8,10,9,11,7,12)(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)(13,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27);(6,11)(10,12)(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=dh,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286502
maximal normal subgroup of	#72286502
maximal quotient of	#72286813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100706);