#185465

order	512
nilpotency class	3
isoclinism	isoclinic class of #185463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239721
maximal normal subgroup of	#72239721
maximal quotient of	#72239731

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92661);