#519027

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331007, #72331014, #72331138, …
maximal normal subgroup of	#72331007, #72331014, #72331138, …
maximal quotient of	#72333188, #72335589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426223);