#6516744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251942, #72255784
maximal normal subgroup of	#72251942, #72255784

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423940);