#188781

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95977);