#355478

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13784
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1472 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)(3,11,13,7)(4,15,16,8)(5,10)(6,9)(12,19)(14,24,27,20)(17,28,29,21)(18,22)(23,25)(26,30)(31,32);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31);(1,2)(3,11,13,7)(4,8)(5,10)(6,9)(12,23,25,19)(14,24,27,20)(15,16)(17,21)(18,22)(26,31,32,30)(28,29);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,15)(10,21)(11,23)(13,25)(14,26)(18,29)(20,30)(22,28)(24,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240437, #72240447, #72240459, …
maximal normal subgroup of	#72240437, #72240447, #72240459, …
maximal quotient of	#72255633, #72262359

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262674);