#279209

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 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)(3,12)(4,9)(5,13)(6,14)(7,10)(8,11)(15,16)(17,18,23,31,22,27,21,26)(19,24,28,32,20,30,29,25);(1,3,6,13,8,4,7,15)(2,5,10,12,11,16,14,9)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,30)(31,32);(1,4,8,3)(2,9,11,12)(5,14,16,10)(6,15,7,13)(17,20,22,19)(18,25,27,32)(21,29,23,28)(24,31,30,26);(1,5,6,12,8,16,7,9)(2,3,10,13,11,4,14,15)(17,21,22,23)(18,26,27,31)(19,28,20,29)(24,32,30,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=hi,d^2=ei,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, 186405);