#6437628

order	512
nilpotency class	3
isoclinism	isoclinic class of #23895
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂	C₂×#2403
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218347, #72218460, #72233452, …
maximal normal subgroup of	#72218347, #72218460, #72233452, …
maximal quotient of	C₂×#6437628, #72238117, #72241280, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344824);