#182229

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1480 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)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(5,6,7,10,8,9,11,12)(13,15)(14,19,21,27)(16,25)(17,24)(18,23)(20,22,26,28);(1,3,4,2)(15,23)(19,27)(22,28)(24,25);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,28,27,22)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279227, #72279276, #72309536
maximal normal subgroup of	#72279227, #72279276, #72309536
maximal quotient of	#72279471

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89425);