#220294

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#99
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#3334	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,6)(3,4)(7,8)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23);(1,2)(3,6)(4,8)(5,7)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(1,3)(4,5)(6,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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.

Relationships

maximal subgroup of	D₈×#410, #72343245, #72343248
maximal normal subgroup of	D₈×#410, #72343245, #72343248

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127490);