#367222

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1219 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)(5,6,11,15)(7,16,22,23)(8,10,14,17)(9,25,28,24)(12,21,18,20)(13,27,30,26)(19,29,34,32)(31,35,36,33);(1,3,4,2)(5,7,17,20,11,22,10,21)(6,12,8,23,15,18,14,16)(9,19,26,35,28,34,27,33)(13,29,24,36,30,32,25,31);(5,8,10,15,11,14,17,6)(7,18,21,16,22,12,20,23)(9,24,27,30,28,25,26,13)(19,32,33,31,34,29,35,36);(5,9,11,28)(6,13,15,30)(7,19,22,34)(8,24,14,25)(10,27,17,26)(12,29,18,32)(16,31,23,36)(20,35,21,33)

Polycyclic presentation:

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

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, 274418);