#7537820

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307362, #72307564, #72307807
maximal normal subgroup of	#72307362, #72307564, #72307807

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7445016);