#181550

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1384 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)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(21,22)(23,31)(24,29)(25,28)(26,30)(27,32)(33,35)(34,36);(6,10,13,12)(11,18,14,19)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,35)(30,36);(5,7)(6,11)(8,16)(9,15)(10,18)(12,19)(13,14)(17,20)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,31,36,35);(7,15)(11,14)(16,20)(18,19)(21,24,26,25)(22,29,30,28)(23,32,33,34)(27,35,36,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88746);