#115763

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266588
maximal normal subgroup of	#72266588
maximal quotient of	#72266589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22959);