#204057

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215872, #72215908, #72215911
maximal normal subgroup of	#72215872, #72215908, #72215911

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111253);