#183648

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dh,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90844);