#6380923

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 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)(5,6)(7,10)(8,9)(11,12)(14,19)(17,26)(20,21)(22,27)(23,25)(24,28);(13,14,18,22)(15,20,16,24)(17,27,26,19)(21,23,28,25);(6,9)(10,12)(13,15,18,16)(14,20,22,24)(17,23,26,25)(19,21,27,28);(13,16,18,15)(14,21,22,28)(17,23,26,25)(19,20,27,24);(5,7,8,11)(6,10,9,12)(14,22)(19,27)(20,24)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250423, #72256939, #72257047, …
maximal normal subgroup of	#72250423, #72256939, #72257047, …
maximal quotient of	#72264943, #72308269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288119);