#497393

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	D₄×#135	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)(8,19,21,35)(9,23)(10,15)(12,20,28,33)(13,17)(14,18)(16,32,22,29)(24,31,36,26)(25,27)(30,34);(1,2)(3,4)(5,7,10,18)(6,11,15,14)(8,16,21,22)(9,17,25,34)(12,26,28,31)(13,27,30,23)(19,32,35,29)(20,24,33,36);(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,29)(14,31)(15,28)(17,33)(18,22)(19,27)(20,34)(23,35)(25,36)(30,32);(2,4)(5,7,10,18)(6,11,15,14)(8,16,21,22)(9,17,25,34)(12,26,28,31)(13,27,30,23)(19,32,35,29)(20,24,33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=h,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	#72286580, #72286595, D₄×#2145, …
maximal normal subgroup of	#72286580, #72286595, D₄×#2145, …
maximal quotient of	#72287091, #72287269, #72287411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404589);