#191535

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 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:

(2,5)(3,7)(6,8)(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);(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);(1,2,3,5,4,6,7,8)(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=g,b^2=e,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224816, #72224817, #72224822, …
maximal normal subgroup of	#72224816, #72224817, #72224822, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98731);