#363204

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#553	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313317, #72313341, #72313349
maximal normal subgroup of	#72313317, #72313341, #72313349

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270400);