#359101

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 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)(5,6,10,15,11,7,16,14)(8,17,20,26,21,12,24,27)(9,18,22,29,23,13,28,30)(19,25,32,36,33,31,34,35);(5,7)(6,11)(8,12)(9,18)(10,14)(13,23)(15,16)(17,21)(19,25)(20,27)(22,29)(24,26)(28,30)(31,33)(32,36)(34,35);(2,4)(5,8,10,20,11,21,16,24)(6,12,14,26,7,17,15,27)(9,19,22,32,23,33,28,34)(13,25,29,35,18,31,30,36);(5,9,10,22,11,23,16,28)(6,13,15,30,7,18,14,29)(8,19,20,32,21,33,24,34)(12,25,27,36,17,31,26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239830, #72252972, #72256869, …
maximal normal subgroup of	#72239830, #72252972, #72256869, …
maximal quotient of	#72308396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266297);