#184932

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

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×#379
prime radical	C₂×#585	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3313 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)(10,15)(12,13)(16,23)(17,20)(18,21)(22,24);(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,24)(20,23);(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280202, #72280211, #72280412
maximal normal subgroup of	#72280202, #72280211, #72280412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92128);