#116121

order	512
nilpotency class	3
isoclinism	isoclinic class of #116120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,f^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fi,d^c=dh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236944, #72236994, #72237000
maximal normal subgroup of	#72236944, #72236994, #72237000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23317);