#181085

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239191, #72239269, #72239287, …
maximal normal subgroup of	#72239191, #72239269, #72239287, …
maximal quotient of	#72239229, #72239594, #72314239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88281);