#147021

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1799
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240706, #72240716, #72240799, …
maximal normal subgroup of	#72240706, #72240716, #72240799, …
maximal quotient of	#72240712, #72240721, #72307758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54217);