#116119

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266588
maximal normal subgroup of	#72266588
maximal quotient of	#72266592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23315);