#229259

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#1667	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 136455);