#376178

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1571 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,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, 283374);