#220808

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#3303	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fi,e^d=eh

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, 128004);