#7514365

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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

3043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257761, #72257767, #72263857, …
maximal normal subgroup of	#72257761, #72257767, #72263857, …
maximal quotient of	#72264482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421561);