#415111

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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)(7,17,20,12)(8,21)(9,13)(10,16)(11,15)(14,23)(18,32,34,35)(19,28,25,26)(22,29)(24,30)(27,33,31,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,28)(14,27)(15,17)(16,26)(21,35)(22,36)(23,33)(24,34)(29,31)(30,32);(2,4)(6,13,15,16)(12,26,17,28)(14,21,29,30)(27,32,31,35);(5,8,10,23,11,24,9,22)(6,14,13,21,15,29,16,30)(7,18,19,33,20,34,25,36)(12,27,28,35,17,31,26,32)

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=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, 322307);