#97569

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,f^c=fg

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257804, #72257971, #72257972
maximal normal subgroup of	#72257804, #72257971, #72257972
maximal quotient of	#72264850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4765);