#441712

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15691
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279719
maximal normal subgroup of	#72279719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348908);