#6346637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346594
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231727, #72249917, C₂×#6346637, …
maximal normal subgroup of	#72231727, #72249917, C₂×#6346637, …
maximal quotient of	#72231710, #72231715, #72249980, …
automorphism group of	#1339, #1340, #1341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253833);