#180965

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2022 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=ci,e^a=eh,g^a=gh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211357, #72212990, #72212999, …
maximal normal subgroup of	#72211357, #72212990, #72212999, …
maximal quotient of	#72322151

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88161);