#375200

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282396);