#420462

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327658);