#212879

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120075);