#116832

order	512
nilpotency class	3
isoclinism	isoclinic class of #10254
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264890, #72265478, #72265483
maximal normal subgroup of	#72264890, #72265478, #72265483

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24028);