#147405

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
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

790 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224893, #72226813, #72226822, …
maximal normal subgroup of	#72224893, #72226813, #72226822, …
maximal quotient of	#72224890

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54601);