#378039

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285235);