#229107

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 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)(3,4)(5,6,8,11,9,7,10,12)(14,18,22,20)(21,24,28,27);(5,7,8,12,9,6,10,11)(13,14,19,18,17,22,15,20)(16,24,26,28,25,27,23,21);(2,4)(13,15,17,19)(14,20,22,18)(16,23,25,26)(21,27,28,24);(13,16,19,26,17,25,15,23)(14,21,20,27,22,28,18,24)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136303);