#376928

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1482 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)(5,6,10,15)(7,12,16,27)(8,20,22,30)(9,11,17,14)(13,24,28,21)(18,32,31,35)(19,29,23,25)(26,34,36,33);(6,11)(7,16)(9,17)(12,25)(13,20)(14,15)(18,31)(21,24)(26,35)(27,29)(28,30)(32,36);(5,7,17,19,10,16,9,23)(6,12,11,25,15,27,14,29)(8,18,21,34,22,31,24,33)(13,26,20,35,28,36,30,32);(1,3,4,2)(5,8)(6,13)(7,18)(9,24)(10,22)(11,20)(12,26)(14,30)(15,28)(16,31)(17,21)(19,34)(23,33)(25,35)(27,36)(29,32)

Polycyclic presentation:

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

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, 284124);