#183224

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1080 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)(6,11)(8,9)(12,18)(13,14)(16,17)(19,20)(21,22,25,29,26,24,28,33)(23,27,32,35,31,30,34,36);(2,4)(5,6,9,14,10,13,8,11)(7,12,17,19,15,20,16,18)(21,23)(22,27)(24,30)(25,32)(26,31)(28,34)(29,35)(33,36);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19)(21,24,25,33,26,22,28,29)(23,30,32,36,31,27,34,35);(6,13)(7,15)(11,14)(16,17)(22,24)(23,31)(29,33)(32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,d^b=dhi,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, 90420);