#358573

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1926 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226691, #72237222, #72237228, …
maximal normal subgroup of	#72226691, #72237222, #72237228, …
maximal quotient of	#72226684, #72239003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265769);