#369195

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 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)(7,16)(8,12)(9,22)(10,17)(11,15)(13,23)(14,25)(18,32)(19,30)(20,33)(21,27)(24,31)(26,29)(28,34)(35,36);(1,3,4,2)(5,7,10,12,11,19,25,27)(6,8,14,16,15,21,17,30)(9,18,23,26,24,33,34,35)(13,20,22,29,28,32,31,36);(5,8,11,21)(6,12)(7,17)(9,20,24,32)(10,16,25,30)(13,26)(14,19)(15,27)(18,31)(22,33)(23,29,34,36)(28,35);(5,9,10,23,11,24,25,34)(6,13,14,22,15,28,17,31)(7,18,12,26,19,33,27,35)(8,20,16,29,21,32,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276391);