#183943

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(21,22,25,29,26,23,30,28)(24,27,32,35,33,31,36,34);(1,2,4,3)(6,10)(8,17)(11,18)(12,13)(14,19)(16,20)(21,23)(22,26)(24,31)(25,28)(27,33)(29,30)(32,34)(35,36);(1,3,4,2)(5,7)(6,11)(8,16)(9,15)(10,18)(12,19)(13,14)(17,20)(22,23)(27,31)(28,29)(34,35);(7,15)(11,14)(16,20)(18,19)(21,24,25,32,26,33,30,36)(22,27,29,35,23,31,28,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eh,c^b=ci,e^b=eh,d^c=dhi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91139);