#231704

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138900);