#186273

order	512
nilpotency class	3
isoclinism	isoclinic class of #186128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62812	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3299 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338224, #72338253, #72338254
maximal normal subgroup of	#72338224, #72338253, #72338254

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93469);