#494068

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239989, #72239996, #72240007, …
maximal normal subgroup of	#72239989, #72239996, #72240007, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401264);