#392199

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299395);