#469909

order	512
nilpotency class	3
isoclinism	isoclinic class of #469875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2607
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16668
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237084, #72237086, #72237096, …
maximal normal subgroup of	#72237084, #72237086, #72237096, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377105);