#193792

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280243, #72302217
maximal normal subgroup of	#72280243, #72302217

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100988);