#386221

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1184 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)(7,9)(8,11)(10,12)(13,14,18,15)(16,23,21,19)(17,25,26,24)(20,27,22,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(5,8,10,9)(6,11,12,7)(13,16,23,14,18,21,19,15)(17,24,27,20,26,25,28,22);(5,9,10,8)(6,11,12,7)(13,17)(14,20)(15,22)(16,24)(18,26)(19,28)(21,25)(23,27)

Polycyclic presentation:

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

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, 293417);