#376182

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14175
prime radical	#23443	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eh,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240432, #72240441, #72240459, …
maximal normal subgroup of	#72240432, #72240441, #72240459, …
maximal quotient of	#72251209, #72251300, #72262382, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283378);