#272856

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

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, 180052);