#146597

order	512
nilpotency class	4
isoclinism	isoclinic class of #12753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53793);