#384567

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei

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, 291763);