#411623

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2304 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2022, #72310626, #72310663
maximal normal subgroup of	D₄×#2022, #72310626, #72310663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318819);