#354030

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3678 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 3678 conjugacy classes of subgroups — too many to draw. Only its 738 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,b^a=be,c^a=cf,e^a=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72267223
maximal subgroup of	#72210949, #72267223, #72267229
maximal normal subgroup of	#72210949, #72267223, #72267229
maximal quotient of	C₄×#10935, #72267446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261226);