#279128

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22,27,30,26,31,25,35)(23,32,33,29,24,28,34,36);(1,2)(3,4)(5,7,9,15,10,16,8,17)(6,11,12,14,13,20,18,19)(21,23,26,24)(22,28,31,32)(25,34,27,33)(29,30,36,35);(2,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(21,24,26,23)(22,29,31,36)(25,34,27,33)(28,30,32,35);(5,7,8,17,10,16,9,15)(6,11,18,19,13,20,12,14)(21,25,26,27)(22,30,31,35)(23,33,24,34)(28,29,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186324);