#358441

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(9,10,14,18,15,19,13,28)(11,20,23,30,24,34,22,16)(12,17,27,32,25,31,26,38)(21,33,37,39,35,40,36,29);(5,6)(7,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,30,28,34)(12,21,26,36,25,35,27,37)(17,29,32,33,31,39,38,40);(1,2,3,4)(6,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,36)(23,37)(24,35)(28,38)(30,39)(34,40);(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320984, #72320992, #72321009
maximal normal subgroup of	#72320984, #72320992, #72321009
maximal quotient of	#72321871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265637);