#184129

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

786 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)(5,6,9,8,10,12,17,16)(7,11,15,14,13,18,20,19)(21,22)(23,27)(24,32)(25,30)(26,29)(28,34)(31,33)(35,36);(2,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,25,32,26,31,35,28)(22,27,30,24,29,33,36,34);(6,12)(7,13)(8,16)(15,20)(21,24,25,33,26,34,35,27)(22,28,30,23,29,32,36,31);(5,8,9,12,10,16,17,6)(7,14,15,18,13,19,20,11)(22,29)(23,31)(28,32)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=chi,d^a=dhi,c^b=ch,d^b=di,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91325);