#375817

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1756 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)(7,15)(8,19)(9,22)(10,14)(11,17)(12,20)(13,24)(16,27)(18,29)(21,30)(23,26)(25,28)(31,32)(33,35)(34,36);(5,7,10,18)(6,11,14,26)(8,16,21,33)(9,23,24,17)(12,27,28,35)(13,29,22,15)(19,31,30,36)(20,34,25,32);(9,24)(11,26)(13,22)(17,23)(19,30)(20,25)(27,35)(32,34);(1,3,4,2)(5,8)(6,12)(7,16)(9,25)(10,21)(11,27)(13,30)(14,28)(15,31)(17,34)(18,33)(19,22)(20,24)(23,32)(26,35)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^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, 283013);