#105750

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

2274 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)(5,6)(7,11,12,9)(8,10)(13,14,16,20,18,22,17,21)(15,23,24,27,25,28,26,19);(5,7)(6,9)(8,12)(10,11)(13,15,17,24,18,25,16,26)(14,19,21,23,22,27,20,28);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238897, #72267927
maximal normal subgroup of	#72238897, #72267927
maximal quotient of	#72268048, #72268049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12946);