#279204

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	#29818	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 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)(17,18,23,26,22,27,21,28)(19,29,30,32,20,24,31,25);(1,2,4,7,5,8,11,9)(3,6,12,10,13,15,14,16)(17,19,22,20)(18,24,27,29)(21,31,23,30)(25,28,32,26);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(1,3,11,14,5,13,4,12)(2,6,9,16,8,15,7,10)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=i,d^2=eh,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, 186400);