#6348382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348364
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,27)(20,26)(21,24)(22,25)(23,29)(28,31)(30,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19,22,28)(18,23,25,32)(20,30,21,29)(24,31,26,27);(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(3,10)(6,15)(9,16)(11,13)(19,28)(23,32)(27,31)(29,30);(2,7)(6,15)(8,12)(9,16)(18,25)(23,32)(24,26)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249743, #72249987
maximal normal subgroup of	#72249743, #72249987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255578);