#146750

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249938, #72249959, #72249982, …
maximal normal subgroup of	#72249938, #72249959, #72249982, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53946);