#410827

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(8,20)(12,25)(16,29)(17,31)(21,24)(26,35)(30,36)(32,34);(5,6,11,15,10,14,9,22)(7,16,23,25,18,29,19,12)(8,13,24,28,20,27,21,33)(17,30,34,35,31,36,32,26);(2,4)(5,7,9,19,10,18,11,23)(6,12,15,16,14,25,22,29)(8,17,21,32,20,31,24,34)(13,26,28,30,27,35,33,36);(5,8)(6,13)(7,17)(9,21)(10,20)(11,24)(12,26)(14,27)(15,28)(16,30)(18,31)(19,32)(22,33)(23,34)(25,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257091, #72257201, #72257214
maximal normal subgroup of	#72257091, #72257201, #72257214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318023);