#231408

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,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, 138604);