#407979

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=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, 315175);