#392350

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(7,17,21,26)(8,22,23,14)(9,12)(10,15)(11,16)(13,20,29,19)(18,32)(24,30,25,27)(28,35)(31,33)(34,36);(6,12,16,15)(8,23)(13,26,29,17)(14,30,22,27)(18,33)(24,25)(28,31,36,32)(34,35);(5,7,9,19,11,21,10,20)(6,13,15,17,16,29,12,26)(8,18,24,34,23,33,25,35)(14,28,30,31,22,36,27,32);(2,4)(5,8)(6,14)(7,18)(9,24)(10,25)(11,23)(12,27)(13,28)(15,30)(16,22)(17,31)(19,34)(20,35)(21,33)(26,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257017, #72257164, #72257207, …
maximal normal subgroup of	#72257017, #72257164, #72257207, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299546);