#207522

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=di,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114718);