#392597

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14693
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215938, #72219682, #72219684, …
maximal normal subgroup of	#72215938, #72219682, #72219684, …
maximal quotient of	#72343137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299793);