#474178

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2607
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16902
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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)(3,13,16,8)(4,17)(5,12)(6,11)(7,10)(9,19)(14,30,31,25)(15,24,22,28)(18,29,32,23)(20,26)(21,27);(1,3,6,15,7,16,5,22)(2,8,11,28,10,13,12,24)(4,18,19,14,21,32,20,31)(9,25,27,29,26,30,17,23);(1,4,5,19,7,21,6,20)(2,9,11,17,10,26,12,27)(3,14,15,18,16,31,22,32)(8,23,24,25,13,29,28,30);(9,26)(17,27)(23,29)(25,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251703, #72251706, #72251711, …
maximal normal subgroup of	#72251703, #72251706, #72251711, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381374);