#121058

order	512
nilpotency class	3
isoclinism	isoclinic class of #10456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#121058, #72240063, #72240064
maximal normal subgroup of	C₂×#121058, #72240063, #72240064
maximal quotient of	C₂×#121058, #72240043, #72240044

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28254);