#374451

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

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

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, 281647);