#497399

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3287	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286595, D₄×#2148, #72309457, …
maximal normal subgroup of	#72286595, D₄×#2148, #72309457, …
maximal quotient of	#72287233, #72287467, #72309534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404595);