#428860

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,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, 336056);