#7522428

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#44927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2356 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)(7,11)(8,10)(9,12)(13,14,18,19)(15,20,16,23)(17,25,27,21)(22,26,24,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,16)(14,20,19,23)(17,26,27,28)(21,22,25,24);(2,4)(13,16,18,15)(14,20,19,23)(17,24,27,22)(21,28,25,26);(2,4)(5,7,9,8)(6,11,12,10)(13,17)(14,21)(15,22)(16,24)(18,27)(19,25)(20,28)(23,26);(5,8,9,7)(6,11,12,10)(13,14,18,19)(15,20,16,23)(17,21,27,25)(22,28,24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283189, #72283348, #72284011, …
maximal normal subgroup of	#72283189, #72283348, #72284011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429624);