#375816

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1732 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)(7,12)(8,19)(9,11)(10,15)(13,20)(14,22)(16,31)(17,26)(18,28)(21,30)(23,25)(24,29)(27,32)(33,35)(34,36);(6,11)(9,22)(12,26)(13,19)(14,15)(17,23)(20,24)(25,28)(27,31)(29,30)(32,34)(35,36);(5,7,10,18)(6,12,15,28)(8,16,21,33)(9,23,22,17)(11,25,14,26)(13,27,29,36)(19,35,30,31)(20,34,24,32);(1,3,4,2)(5,8)(6,13)(7,16)(9,24)(10,21)(11,19)(12,27)(14,30)(15,29)(17,34)(18,33)(20,22)(23,32)(25,35)(26,31)(28,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,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,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, 283012);