#270599

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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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)(5,6,10,14)(7,12,17,20)(8,13,9,11)(15,19,16,18)(23,29)(26,35)(30,36)(31,34);(6,11)(8,9)(12,18)(13,14)(15,16)(19,20)(21,22,24,27)(23,30,29,36)(25,33,32,28)(26,34,35,31);(2,4)(5,7,10,17)(6,12,14,20)(8,15,9,16)(11,18,13,19)(21,23,24,29)(22,26,27,35)(25,34,32,31)(28,36,33,30);(5,8)(7,15)(9,10)(11,13)(16,17)(18,19)(23,29)(26,35)(30,36)(31,34)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,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, 177795);