#175237

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279315
maximal normal subgroup of	#72279315
maximal quotient of	#72279383, #72279393

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 82433);