#368900

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14023
prime radical	#60469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215938, #72219683, #72219685
maximal normal subgroup of	#72215938, #72219683, #72219685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276096);