#434427

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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:

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

Polycyclic presentation:

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

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, 341623);