#183487

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

730 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,c^a=ci,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, 90683);