#7532104

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256840, #72256891, #72256980, …
maximal normal subgroup of	#72256840, #72256891, #72256980, …
maximal quotient of	#72307882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439300);