#354054

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4018 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241072, #72241086, #72265217, …
maximal normal subgroup of	#72241072, #72241086, #72265217, …
maximal quotient of	#72266011, #72285516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261250);