#272396

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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)(3,11)(4,7)(5,15)(6,8)(9,13)(10,12)(14,16)(17,18,22,24,20,25,21,26)(19,27,30,31,29,23,28,32);(1,3,6,13)(2,7,8,16)(4,5,14,12)(9,15,11,10)(17,19)(18,23)(20,29)(21,28)(22,30)(24,32)(25,27)(26,31);(2,8)(7,16)(9,11)(10,15)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,4,6,14)(2,9,8,11)(3,12,13,5)(7,10,16,15)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179592);