#6347408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347398
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272561, #72272573
maximal normal subgroup of	#72272561, #72272573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254604);