#183137

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,4,3)(6,10)(7,8)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(5,6,8,12,9,11,7,10)(14,19,22,21)(15,23)(20,27,28,26)(24,25);(1,3,4,2)(6,11)(10,12)(13,15)(14,20)(16,25)(17,24)(18,23)(19,26)(21,27)(22,28);(5,7,9,8)(6,10,11,12)(13,16,18,17)(14,21,22,19)(15,24,23,25)(20,26,28,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286392, #72286485, #72320007, …
maximal normal subgroup of	#72286392, #72286485, #72320007, …
maximal quotient of	#72321464, #72321595, #72321686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90333);