#270611

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177807);