#273011

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180207);