#278622

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185818);