#118005

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257811, #72257884, #72257885, …
maximal normal subgroup of	#72257811, #72257884, #72257885, …
maximal quotient of	#72264800, #72265380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25201);