#6372398

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239856, #72239869, #72239880, …
maximal normal subgroup of	#72239856, #72239869, #72239880, …
maximal quotient of	#72239862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279594);