#285127

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,21,25,22,20,31,24)(19,26,29,27,30,32,28,23);(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(17,19,21,29,22,30,31,28)(18,23,25,26,20,27,24,32);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,20)(18,22)(19,27)(21,25)(23,30)(24,31)(26,29)(28,32);(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(17,20)(18,22)(19,27)(21,24)(23,30)(25,31)(26,28)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192323);