#227599

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#18480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287015, #72322056, #72322077
maximal normal subgroup of	#72287015, #72322056, #72322077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134795);