#204557

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1260 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)(5,6,7,11,8,10,12,9)(14,18,21,24)(20,25,27,28);(6,9)(7,12)(10,11)(13,14,19,18,17,21,16,24)(15,20,26,25,22,27,23,28);(1,2,3,4)(6,10)(9,11)(13,15,17,22)(14,20,21,27)(16,23,19,26)(18,25,24,28);(15,22)(20,27)(23,26)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319882, #72320091, #72320224
maximal normal subgroup of	#72319882, #72320091, #72320224
maximal quotient of	#72321595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111753);