#7517231

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514060
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×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,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,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257270, #72257505, #72257761, …
maximal normal subgroup of	#72257270, #72257505, #72257761, …
maximal quotient of	#72264366

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424427);