#229300

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309438, #72309444, #72309525, …
maximal normal subgroup of	#72309438, #72309444, #72309525, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136496);