#147691

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321310, #72321352, #72321415, …
maximal normal subgroup of	#72321310, #72321352, #72321415, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54887);