#519013

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×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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	C₄×#36096, #72306249, #72306264
maximal normal subgroup of	C₄×#36096, #72306249, #72306264

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426209);