#354095

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224705, #72226691
maximal normal subgroup of	#72224705, #72226691
maximal quotient of	#72224696, #72224699

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261291);