#6380419

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264011, #72264037, #72308773, …
maximal normal subgroup of	#72264011, #72264037, #72308773, …
maximal quotient of	#72264907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287615);