#7192958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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
prime radical	#24152	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257122, #72257132, #72257181, …
maximal normal subgroup of	#72257122, #72257132, #72257181, …
maximal quotient of	#72265389

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100154);