#6451782

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2404
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3837 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,29)(16,23)(17,24)(18,32)(19,26)(20,28)(21,27)(22,30)(25,31);(1,3)(2,8)(4,18)(5,16)(6,14)(7,17)(9,25)(10,13)(11,24)(12,23)(15,20)(19,31)(21,30)(22,26)(27,29)(28,32);(3,14,17,16)(8,13,23,24)(15,30,31,18)(22,29,32,25);(1,4)(2,9)(3,15,17,31)(5,19)(6,20)(7,21)(8,22,23,32)(10,26)(11,28)(12,27)(13,29,24,25)(14,30,16,18);(1,5,7,6)(2,10,12,11)(3,16,17,14)(4,19,21,20)(8,13,23,24)(9,26,27,28)(15,18,31,30)(22,29,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249649, #72261154, #72263053, …
maximal normal subgroup of	#72249649, #72261154, #72263053, …
maximal quotient of	#72256011, #72262282, #72262980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358978);