#519042

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=g,e^2=f,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	#72297725, #72297729, #72297774
maximal normal subgroup of	#72297725, #72297729, #72297774
maximal quotient of	#72298564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426238);