#114384

order	512
nilpotency class	3
isoclinism	isoclinic class of #99923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240315, #72240394, #72240395, …
maximal normal subgroup of	#72240315, #72240394, #72240395, …
maximal quotient of	#72261280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21580);