#227113

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134309);