#434428

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(7,14)(8,13,24,31)(9,25)(10,26)(11,16)(12,18)(15,21)(17,29,20,34)(19,32)(22,28,33,27)(23,36,35,30);(2,3)(5,7,11,19)(6,12,16,25)(8,20,24,17)(9,21,18,10)(13,28,31,27)(14,15,32,26)(22,23,33,35)(29,36,34,30);(1,2,4,3)(5,8,21,35,11,24,10,23)(6,13,15,30,16,31,26,36)(7,17,9,22,19,20,18,33)(12,27,14,29,25,28,32,34);(1,3,4,2)(5,9,10,7,11,18,21,19)(6,14,15,25,16,32,26,12)(8,22,23,17,24,33,35,20)(13,29,30,28,31,34,36,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ef,d^2=e,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, 341624);