#376777

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(5,6,12,14)(7,15,18,19)(8,10,20,11)(9,16,13,17);(5,7)(6,13)(8,19)(9,14)(10,17)(11,16)(12,18)(15,20);(5,8,10,14,12,20,11,6)(7,9,17,15,18,13,16,19);(1,3,4,2)(5,9,12,13)(6,7,14,18)(8,17,20,16)(10,15,11,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312223, #72312295
maximal normal subgroup of	C₄×#24472, #72312223, #72312295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283973);