#395223

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
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	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1809 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)(7,13)(8,20)(9,15)(10,27)(11,16)(12,17)(14,23)(18,35)(19,30)(21,28)(22,31)(24,33)(25,34)(26,29)(32,36);(1,2,4,3)(5,7,12,9)(6,13,17,15)(8,21,25,36)(10,19,11,26)(14,31,33,35)(16,29,27,30)(18,23,22,24)(20,28,34,32);(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,25)(13,28)(15,32)(16,34)(17,33)(19,36)(20,27)(21,26)(29,31)(30,35);(1,3,4,2)(5,9,12,7)(6,15,17,13)(8,22,25,18)(10,19,11,26)(14,32,33,28)(16,29,27,30)(20,35,34,31)(21,24,36,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302419);