#6383748

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239439, #72257326, #72257371
maximal normal subgroup of	#72239439, #72257326, #72257371
maximal quotient of	#72239405

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290944);