#184126

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,d^b=di,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343922, #72343928, #72343974
maximal normal subgroup of	#72343922, #72343928, #72343974
maximal quotient of	#72344004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91322);