#6384392

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308624, #72308792, #72308834
maximal normal subgroup of	#72308624, #72308792, #72308834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291588);