#220296

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#99
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127492);