#117748

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

788 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)(7,10)(11,12)(13,14,17,20,18,21,16,24)(15,19,22,28,23,27,25,26);(5,6,10,12,8,11,7,9)(15,22,23,25)(19,26,27,28);(2,4)(13,15)(14,19)(16,25)(17,22)(18,23)(20,28)(21,27)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215279, #72216008, #72216015
maximal normal subgroup of	#72215279, #72216008, #72216015

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24944);