#183476

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,d^b=di,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285487, #72286485, #72319606, …
maximal normal subgroup of	#72285487, #72286485, #72319606, …
maximal quotient of	#72285908, #72321675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90672);