#497978

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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)(5,6,11,13,10,16,21,29,12,17,20,30,9,15,7,18)(8,22,27,34,26,14,23,28,24,33,35,36,25,32,19,31);(1,2)(3,4)(5,7,10,11,12,21,9,20)(6,13,15,18,17,30,16,29)(8,23,26,35,24,19,25,27)(14,31,22,36,32,28,33,34);(5,8,10,26,12,24,9,25)(6,14,15,22,17,32,16,33)(7,19,11,27,21,23,20,35)(13,28,18,34,30,31,29,36);(2,4)(8,24)(14,32)(19,23)(22,33)(25,26)(27,35)(28,31)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi,d^c=di

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, 405174);