#279198

order	512
nilpotency class	3
isoclinism	isoclinic class of #279179
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	#19446	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186394);