#181798

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(17,18,21,25,22,24,20,29)(19,26,28,23,27,32,30,31);(1,3,5,14,6,12,4,13)(2,7,10,16,8,15,9,11)(17,19)(18,23)(20,30)(21,28)(22,27)(24,31)(25,32)(26,29);(3,12)(7,15)(11,16)(13,14)(19,27)(23,31)(26,32)(28,30);(2,8)(7,15)(9,10)(11,16)(18,24)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,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	#72238984, #72238987, #72238995, …
maximal normal subgroup of	#72238984, #72238987, #72238995, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88994);