#359574

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

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×#3586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3745 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,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, 266770);