#674174

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

5950 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2409, #72264077, #72264098
maximal normal subgroup of	D₄×#2409, #72264077, #72264098

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 581370);