#219455

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23798	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(10,13)(11,14)(15,16)(17,18,21,25,22,26,30,24)(19,23,28,32,27,31,20,29);(1,3,6,2)(4,7,5,8)(17,19,21,28,22,27,30,20)(18,23,25,32,26,31,24,29);(1,4,6,5)(2,7,3,8)(9,10,14,16,12,15,11,13)(17,20,30,27,22,28,21,19)(18,23,25,32,26,31,24,29);(1,5,6,4)(2,7,3,8)(19,27)(20,28)(23,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343125, #72343139, #72343171
maximal normal subgroup of	#72343125, #72343139, #72343171
maximal quotient of	#72343535, #72343545

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126651);