#426740

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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)(3,4)(5,6,11,15,10,14,17,25)(7,16,20,32,19,9,13,26)(8,21,23,30,24,31,27,12)(18,33,29,22,34,36,35,28);(5,7,17,13,10,19,11,20)(6,9,25,32,14,16,15,26)(8,18,23,29,24,34,27,35)(12,28,21,33,30,22,31,36);(5,8,17,23,10,24,11,27)(6,12,25,21,14,30,15,31)(7,18,13,29,19,34,20,35)(9,22,32,36,16,28,26,33);(2,4)(5,9)(6,13)(7,15)(8,22)(10,16)(11,26)(12,29)(14,20)(17,32)(18,31)(19,25)(21,34)(23,36)(24,28)(27,33)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333936);