#365446

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#13975
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1417 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)(3,14,19,26,18,28,17,8)(4,9,20,22,10,5,21,23)(15,31,32,16,25,29,30,27);(1,3,6,19,7,18,24,17)(2,8,12,14,13,26,11,28)(4,15,20,32,10,25,21,30)(5,16,23,29,9,27,22,31);(1,4)(2,9)(3,15,18,25)(5,13)(6,20)(7,10)(8,16,26,27)(11,23)(12,22)(14,29,28,31)(17,30,19,32)(21,24);(1,5)(2,10)(3,16)(4,13)(6,23)(7,9)(8,25)(11,20)(12,21)(14,30)(15,26)(17,31)(18,27)(19,29)(22,24)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319834, #72342962, #72342973
maximal normal subgroup of	#72319834, #72342962, #72342973
maximal quotient of	#72343078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272642);