#376929

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

1468 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)(8,21)(9,22)(10,24)(11,15)(12,19)(13,23)(14,28)(17,35)(18,30)(20,32)(25,31)(26,27)(29,36)(33,34);(5,7,11,20)(6,12,15,31)(8,17,22,34)(9,18,21,29)(10,25,28,19)(13,30,26,36)(14,32,24,16)(23,35,27,33);(5,8,10,26,11,22,28,13)(6,9,14,27,15,21,24,23)(7,17,19,30,20,34,25,36)(12,29,16,33,31,18,32,35);(1,3,4,2)(5,9)(6,13)(7,18)(8,14)(10,27)(11,21)(12,30)(15,26)(16,34)(17,32)(19,35)(20,29)(22,24)(23,28)(25,33)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=h,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, 284125);