#148345

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250515, #72250522, #72250536, …
maximal normal subgroup of	#72250515, #72250522, #72250536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55541);