#497604

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×Q₈² ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

776 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)(6,7)(9,10,14,19,15,20,12,24)(11,16,21,31,23,32,25,39)(13,18,26,34,28,35,27,17)(22,30,37,29,38,40,36,33);(1,2,4,5)(3,7,8,6)(9,11)(10,16)(12,25)(13,22)(14,21)(15,23)(17,33)(18,30)(19,31)(20,32)(24,39)(26,37)(27,36)(28,38)(29,34)(35,40);(9,12,15,14)(10,17,20,34)(11,21,23,25)(13,27,28,26)(16,29,32,33)(18,19,35,24)(22,37,38,36)(30,39,40,31);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271508
maximal normal subgroup of	#72271508
maximal quotient of	#72271533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404800);