#183978

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91174);