#181772

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1712 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,25)(24,28)(26,27);(6,10)(7,8)(11,12)(13,15,17,26,18,24,16,25)(14,19,22,28,20,27,21,23);(5,7,9,8)(6,11,12,10)(15,24)(19,27)(23,28)(25,26);(6,12)(10,11)(14,20)(19,27)(21,22)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=h,g^2=hi,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257421, #72257427, #72257446, …
maximal normal subgroup of	#72257421, #72257427, #72257446, …
maximal quotient of	#72257686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88968);