#270612

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=i,d^2=h,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, 177808);