#388330

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×C₄×D₄
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1339 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=eh,e^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295526);